Position an oven rack in the lower third of the oven (I use the rack position just below center). Preheat the oven to 400°F. Use an ungreased nonstick muffin pan (standard size, not mini). If your pan isn’t nonstick, you can lightly grease the wells as extra insurance against sticking.
Allow the pie crusts to soften at room temperature according to the package directions, then unroll on a lightly floured board. Use a 3½-inch round cutter or the rim of a drinking glass to cut 12 circles from the pastry, 6 from each crust.
Gently ease a pastry circle into each cup of a standard 12-cup muffin pan, pressing it against the bottom and sides without stretching it.
Peel, core, and dice the apples into approximately ¼-inch pieces. Measure out 3 cups and place them in a large bowl. Toss with the lemon juice and vanilla extract.
In a small bowl, combine the brown sugar, flour, cinnamon, nutmeg, and kosher salt. Add to the apples and toss until evenly coated. Set the bowl aside while you make the crumb topping.
In a medium bowl, combine the flour, brown sugar, cinnamon, and salt. Add the softened butter and cut it into the dry ingredients with a pastry blender or the tines of a fork until well incorporated. Use your fingers to work the mixture until it holds together in clumps, then break it into coarse crumbs.
Divide the apple mixture evenly among the pastry-lined cups, gently settling the pieces into place, about ¼ cup per pie. Divide any juices remaining in the bowl among the pies.
Divide the topping evenly among the pies, about 1½ tablespoons per pie. Spoon it onto the center of each pie, letting the crumbs fall outward to cover the filling, keeping the topping inside the pastry edges. Brush any stray crumbs off the surface of the muffin pan before baking.
Bake for 18 minutes, then check the pies. If the crumb topping is golden, loosely drape a sheet of foil over the pan. Continue baking for an additional 5 to 7 minutes or until the pastry is golden, the filling is bubbling, and the apples are tender when pierced with a toothpick or thin knife.
Let cool in the pan for 5 to 10 minutes, then carefully loosen the edges with a thin knife or the tines of a fork and transfer the pies to a wire rack. Serve warm or at room temperature.