Preheat your oven to 375°F (190°C). Lightly spray a standard 12-cup muffin tin with non-stick baking spray.
In a medium skillet over medium heat, melt the 1 tablespoon of unsalted butter. Add the finely diced apples, brown sugar, and cinnamon.
Sauté the mixture for 5-7 minutes, stirring gently until the apples are tender and a syrup forms. Remove from heat and let the filling cool completely.
Unroll two of the pie crusts. Using a 3.5-inch round cookie cutter, stamp out 12 circles of dough. Gently press one circle into each cup of the muffin tin, molding it up the sides.
Spoon a generous tablespoon of the cooled, spiced apple filling into each pie crust cup.
Unroll the remaining two pie crusts and slice them into thin strips. Weave the strips over the top of each mini pie to create a classic lattice pattern. Press the edges down to seal.
Using a pastry brush, lightly brush the beaten egg over the tops of the lattice crusts.
Bake on the center rack for 18-20 minutes until the crusts are a deep golden brown and the filling is bubbling.
Let the pies cool in the tin for at least 10-15 minutes before carefully popping them out to serve.