Preheat oven to 350°F (175°C). Line a baking sheet with parchment.
In a large bowl, whisk browned butter with both sugars until smooth. Add vanilla and eggs, whisk until glossy.
Fold in flour, baking soda, baking powder, salt, cinnamon, and nutmeg until a thick dough forms.
Scoop 2-inch dough balls onto baking sheet, spacing 2 inches apart.
Bake 12 minutes, until edges are set and tops crinkle. Let cool.
For filling: rinse apple pie filling, place apples in saucepan. Add sugars, cinnamon, nutmeg, lemon juice, vanilla, cornstarch, and water. Cook over medium heat until softened and thickened. Let cool.
For caramel: cook sugar and water in saucepan until amber. Lower heat, whisk in butter and cream. Stir until smooth, then add salt.
Spoon apple filling into center of cookies. Drizzle with caramel. Serve warm or cooled.