Make the dough
In a small saucepan, heat the milk until lukewarm (about 95°F). Remove from the heat and whisk in the yeast until it dissolves. Add the oil and butter—the butter may begin to melt, but it’s OK if it doesn’t melt completely—and then whisk in the sugar. Let rest until the yeast just begins to float to the surface, about 5 minutes.
In a stand mixer fitted with the paddle attachment (or in a large bowl), combine the flour, salt, and egg. Add the yeast mixture and mix on low speed (or with a large spoon) until a coarse ball of dough forms, about 1 minute. Let rest for 5 minutes.
Replace the paddle attachment with the dough hook and mix on medium-low speed (or knead by hand on a lightly oiled work surface) until the dough feels soft, supple, and pliable, about 3 minutes; it should feel tacky to the touch, but not sticky, and pull away from your finger when poked instead of sticking to it. If the dough is too sticky, add 1 Tbs. flour at a time, kneading to incorporate. If it’s stiff, knead in 1 Tbs. of milk at a time.
Rub a little vegetable oil on a work surface to create an 8-inch circle and put the dough on this spot. Stretch and fold the dough over itself from all four sides to the center, crimping it where the folded ends meet, to form it into a tight, round ball.
Put the dough seam side down in a lightly oiled bowl that’s twice the size of the dough. Tightly cover with plastic wrap. Let sit at room temperature until doubled in size, about 90 minutes.
Shape the rolls
Mist two 12-cup muffin tins with vegetable oil spray.
With a bench knife, divide the dough into 24 pieces, about 2 oz. each. Wipe the work surface with a damp towel or mist lightly with vegetable oil spray (don’t use flour). Form your hand into a cup by curling your fingers slightly, with your thumb snug against your index finger. Use this cupped hand to press and rotate a piece of dough as if you were trying to screw it through the dampened work surface. It should form a smooth round ball within 5 seconds and pop up into your cupped hand.
Using kitchen shears, cut through the top center of the roll, almost all the way through, leaving about 1/4-inch of dough at the bottom to hold the two halves together. Cut one of the halves twice more at 1/2-inch intervals parallel to the first cut. Repeat these cuts on the other half for a total of six sections, each about 1/2-inch wide. Place the dough, cut side up, into the prepared muffin pan. Brush the melted butter over the top and down into the cuts. Repeat with the remaining dough, placing 12 rolls into each muffin pan.
Mist the top of the rolls with vegetable oil spray and cover the pans loosely with plastic wrap. Allow the rolls to proof at room temperature until they nearly double in size, 60 to 90 minutes. The time vary depending on the temperature of the room and the dough; if you made the dough ahead and refrigerated it, the cold dough will take longer to proof.
Position the oven racks in the upper and lower thirds of the oven. If using a convection oven, heat the oven to 350°F; if using a conventional oven, heat the oven to 375°F. Bake the rolls for 7 minutes. Rotate the tins 180 degrees and swap their placement on the racks. Continue baking until the rolls turn rich golden-brown on top, 7 to 9 minutes more. Let the rolls cool for 5 minutes before removing them from the tin.
Make Ahead Tips
The dough may be made up to 4 days ahead of shaping. Kept wrapped in plastic and refrigerated, it will slowly rise to double its size. Shape the dough right out of the fridge. Cold dough will take longer to proof than room temperature dough.