Cook 3/4 cup butter in a large skillet over medium-low heat until melted and light brown, 12 to 15 minutes. Keep an eye on it; it can burn. Pour into a bowl. Freeze until solid, about 30 minutes.
Meanwhile, for the filling melt 2 tablespoons butter in the same skillet. Add apples, 2 tablespoons brown sugar, 1 teaspoon cinnamon, lemon juice, nutmeg, and pinch of salt. Cook, stirring occasionally, until apples are golden brown and caramelized, about 15 minutes. Remove from heat. Add 3 tablespoons brown sugar; toss to coat.
Preheat the oven to 375 degrees F (190 degrees C). Line a 9-inch square baking pan with enough parchment paper to create an overhang on all sides.
Stir together flour, oats, 2/3 cup brown sugar, baking powder, 1/4 teaspoon cinnamon, and 1/4 teaspoon salt in a bowl. Coarsely shred frozen browned butter or chop into small pieces. Cut butter into flour mixture until mixture resembles coarse meal. Add egg yolk and vanilla; stir with a fork just until a crumbly dough comes together. Press about two-thirds of the dough (2 2/3 cups) in an even layer in the prepared baking pan.
Bake until lightly browned, about 10 minutes.
Spread apple mixture over the crust. Crumble remaining dough over top. Bake until golden brown, about 25 minutes. Let cool completely on a wire rack. Using parchment sides to lift bars, transfer to a cutting board and cut into squares.