Preparation

Make the dough:

• Cut the butter into 3/4-inch cubes. Wrap them in plastic and freeze until hard, at least 30 minutes. Put the all-purpose flour, cake flour, salt, and baking powder in a metal bowl and freeze for at least 30 minutes.
• Put the cold flour mixture in a food processor and process for a few seconds to combine.
• Cut the cold cream cheese into three or four pieces and add it to the flour mixture. Process for 20 seconds (the mixture should resemble fine meal). Add the frozen butter cubes and pulse until none of the butter pieces is larger than a pea, about five 3-second pulses. (Toss with a fork to see it better.)
• Add the cream and vinegar and pulse in short bursts until the dough starts to come together (which will take a minute or two); the dough will still look crumbly but if you press it between your fingers, it should become smooth. Turn it out onto a clean work surface. Gather and press the dough together to form a unified mass.
  
• Cut the dough in half and put each half on a large piece of plastic wrap. Loosely cover the dough with the plastic. Using the wrap as an aid (to avoid warming the dough with your bare hands), shape one half of the dough into a flat disk and the other into a flat rectangle. Wrap each tightly in the plastic and refrigerate for at least 45 minutes and up to 24 hours.

Roll out the bottom crust:

• Remove the disk of dough from the fridge (keep the rectangle refrigerated); if it’s very firm, let it sit at room temperature until it’s pliable enough to roll, 10 to 15 minutes.
• Set the dough between two sheets of plastic wrap sprinkled lightly with flour. Roll it out to a 13-inch round that’s 1/8 inch thick, occasionally loosening and reapplying the plastic wrap.
• Remove one piece of plastic and flip the dough into a standard metal 9-inch pie pan (it should be 1-1/4 inches deep and hold 4 cups of liquid). Fit the dough into the pan and carefully peel off the plastic. Trim the dough so there’s a 3/4-inch overhang. Fold the overhang underneath itself to create an edge that extends about 1/4 inch beyond the rim of the pie pan. Cover the dough-lined pie plate with plastic wrap and refrigerate.

Make the filling and top the pie:

• Peel the peaches. Halve each peach, remove the pit, and slice each half into eight thin wedges; you should have 6 cups.
• Put the peaches in a large bowl and sprinkle the lemon juice over them. Sprinkle on the sugar and salt and toss gently to mix. Let sit at room temperature for at least 30 minutes and up to 12 hours. Transfer them to a colander suspended over a bowl to collect the juices; you should have almost 1 cup of liquid (if the peaches sat for several hours, you’ll have 1 to 1-1/2 cups liquid).
• Pour the juices into a small, nonstick saucepan set over medium heat. Boil down the liquid, swirling but not stirring, until it’s syrupy, about 10 minutes; it should reduce to 1/3 to 1/2 cup, depending on how much liquid you started with. Set aside to cool for 1 or 2 minutes.
• Meanwhile, transfer the peaches to a bowl and toss them with the cornstarch and almond extract until all traces of cornstarch have disappeared. Pour the reduced peach juices over the peaches, tossing gently. (Don’t worry if the liquid hardens on the peaches; it will dissolve during baking.)
• Remove the rectangle of dough from the refrigerator and let it sit at room temperature until it’s pliable enough to roll, 10 to 15 minutes. Roll the dough on a lightly floured surface to an 11×14-inch or larger rectangle (if it becomes an oval, that’s fine); it should be no more than 1/8 inch thick.
• Cut ten 3/4-inch-wide strips lengthwise down the rectangle, using a ruler to measure and mark 3/4-inch intervals and to cut a straight edge. If you want a crimped edge on the strips, use a fluted pastry wheel.
• Stir the peach filling a few times and scrape it into the pie shell. Arrange five strips of dough evenly over the filling, starting with a long strip for the center. Gently fold back every other strip (the second and the fourth) to a little past the center. Choose another long strip of dough, hold it perpendicular to the other strips, and set it across the center of the pie.
  
• Unfold the two folded strips so they lie flat on top of the perpendicular strip.  Now fold back the strips that weren’t folded back last time (the first, third, and fifth ones).
  
• Lay a second perpendicular strip of dough about 3/4-inch away from the last one.  Unfold the three folded strips.  Fold back the orginal two strips, set a a third perpendicular strip of dough 3/4 inch from the last one, and unfold the two strips.
  
• Repeat on the other side with the two remaining strips: fold back alternating strips, lay a strip of dough on top, and unfold. Remember to alternate the strips that are folded back to create a woven effect. Trim the strips to a 1/2-inch overhang. Moisten the underside of each one and tuck it under the bottom crust, pressing to make it adhere.  Crimp or flute the edges if you like. 
  

Bake and let the pie cool:

• Lightly cover the assembled pie with plastic wrap and refrigerate for 1 hour. After 30 minutes of chilling, set an oven rack on the lowest rung and put a foil-lined baking stone or baking sheet on it. Heat the oven to 425°F.
• When the pie has chilled for 1 hour, brush the lattice with the milk and sprinkle on the sugar.
• Set the pie directly on the baking stone or sheet. Bake until the juices are bubbling all over (the bubbles should be thick and slow near the pan edges), 40 to 50 minutes. After the first 15 minutes, cover the rim with foil or a pie shield. If the lattice starts to darken too much in the last 10 minutes of baking, cover it loosely with a piece of foil that has a vent hole poked in the center.
• Let the pie cool on a rack until the juices have thickened, 3 hours.