Preheat your oven to 350°F (175°C). Grease and line an 8x8-inch square baking pan with parchment paper, leaving some overhang on two sides for easy removal.
In a bowl mix the cubed cold butter, brown sugar, salt, flour, oats, and cinnamon. Use either your hands or a pastry cutter. Once the mixture is combined well and has small clumps, set aside in the fridge for later.
1/2 cup all-purpose flour, 1/4 cup oats, 1/3 cup brown sugar, 1/2 teaspoon cinnamon, 1/2 teaspoon salt, 1/4 cup butter
Make the crust
In a medium bowl, mix the butter, flour, both sugars, vanilla extract, and salt together until soft cookie dough forms.
1/4 cup granulated sugar, 1 tbsp brown sugar, 1/2 tsp vanilla extract, 1/4 tsp salt, 1/2 cup butter, 1 cup all purpose flour
Spread the dough on an 8x8-inch pan lined with parchment paper. Press to form an even layer. Bake in the oven for 15 minutes.
Make the apple mixture
In a large bowl, toss the sliced apples with ground cinnamon, granulated sugar, nutmeg, and lemon juice. Spread the apple mixture evenly over the shortbread crust.