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.
Make the lattice top:
Remove the rectangle of dough from the refrigerator; if it’s very firm, let it sit at room temperature until it’s pliable enough to roll, 10 to 15 minutes. Roll it out between two sheets of parchment to a rectangle slightly larger than 14x9 inches. Remove the top sheet of parchment. Trim the dough to an exact 14x9-inch rectangle. Cut 12 strips that are 14 inches long and 3/4 inch wide. If the dough gets soft, slide the parchment and dough onto a baking sheet and chill briefly before continuing.
On a parchment-lined baking sheet, arrange 6 strips horizontally, setting them 3/4 inch apart; these will be the "bottom" strips (the other 6 will be your "top" strips). Fold back every other bottom strip halfway, starting with the strip closest to you. Lay one top strips vertically, slightly right of center.
Unfold the folded strips and fold back the other three strips. Lay a second top strip 3/4 inch to the left of the first. Now fold back alternating strips on the right, starting at the top. Lay another top strip 3/4 inch to the right of the center strip; unfold the folded strips. Repeat left and right with the rest of the strips.
Dab a little water between the strips where they overlap, pressing gently to seal. Cover the lattice loosely with plastic and put the baking sheet in the fridge while you make your filling and roll out the bottom crust.
Assemble the pie:
Remove the disk of dough from the fridge; if it’s very firm, let it sit at room temperature until it’s pliable enough to roll, 10 to 15 minutes.
Toss all the filling ingredients together in a large bowl until well blended.
Set the dough between two sheets of parchment. Roll it out to a 13-inch round that’s 1/8 inch thick, occasionally loosening and reapplying the parchment.
Remove one piece of parchment and flip the dough into a standard glass 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.
Pile the filling into the bottom crust. Remove the lattice top from the fridge and put your palm under the parchment at the center of the lattice. Lift the paper and invert the lattice onto the filling.
Trim the crust, leaving a 1/2-inch margin from the edge of the pie plate. Press the edges together, fold them under, and use your thumb and index fingers to pinch-crimp a fluted edge.
Bake the pie:
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.
Photo: Scott Phillips