Preheat the oven to 350°F (175°C). Melt the 4 tablespoons of salted butter in a large skillet over medium-low heat. Once fully melted, remove the skillet from the heat and transfer 3 tablespoons of the butter to a separate bowl. Keep the remaining 1 tablespoon of melted butter in the skillet for cooking the apples.
Peel the 3 apples and cut them into approximately 1/2-inch cubes. Place the diced fruit in a bowl and toss thoroughly with 1 teaspoon lemon juice until evenly coated.
Return the skillet containing the reserved tablespoon of butter to medium-low heat. Add the lemon-coated apples, followed by the packed light brown sugar, granulated sugar, vanilla extract, cinnamon, and nutmeg.
Cook for 5–6 minutes, stirring occasionally, until the apples have softened slightly and the mixture is fragrant. Allowing the apples to remain undisturbed for an additional minute can encourage a little more caramelization without altering the remaining preparation.
Whisk together 1 teaspoon cornstarch and 1 tablespoon water in a small bowl until completely smooth.
Pour the cornstarch mixture into the cooked apples and stir thoroughly. Continue cooking for approximately 2 minutes, until the released juices form a thick, sauce-like coating around the fruit.
Remove the skillet from the heat and allow the apple mixture to cool slightly before assembling the turnovers.
Place one thawed phyllo sheet on a clean work surface and lightly brush it with some of the 3 tablespoons of reserved melted butter. Position another sheet on top and brush again. Continue until four phyllo sheets have been stacked and buttered.
Repeat the same procedure with the remaining phyllo until three separate stacks of four layers each have been prepared.
Using a sharp knife or pizza cutter, divide each four-layer stack in half to create two rectangles, producing six portions in total.
Spoon several tablespoons of the cooled apple filling near the upper-left area of each phyllo rectangle.
Fold the remaining pastry over the filling. Bring the top edge toward you, then fold the ends over and underneath so the turnover is enclosed and sealed.
Arrange each turnover seam-side down on the prepared baking sheet. Brush the surfaces with the remaining melted butter.
Bake at 350°F (175°C) for approximately 15 minutes, or until the phyllo is crisp and evenly golden. Monitor the pastries carefully during the final few minutes because the thin phyllo layers can brown rapidly.
Allow the baked turnovers to rest on the baking sheet for 5–10 minutes.
Serve while still warm. If desired, finish with a light coating of sifted powdered sugar and a drizzle of caramel sauce. Vanilla ice cream may also be served alongside each turnover.