Peel, core, and finely dice the apples into small, uniform pieces. In a medium saucepan or skillet, combine the diced apples, brown sugar, butter, cinnamon, and nutmeg over medium heat.
Cook for 5 to 7 minutes, stirring occasionally, until the apples soften slightly but still hold their shape.
Whisk the cornstarch with the water and lemon juice until smooth, then pour it into the apple mixture. Cook for another 2 to 3 minutes until the filling thickens into a glossy, jammy consistency. Remove from heat and cool completely.
In a large mixing bowl, beat the softened butter and brown sugar together on medium speed until light and fluffy, about 2 minutes. Add the egg yolk and vanilla extract, then mix until fully combined.
In a separate bowl, whisk together the flour, baking powder, and salt. Gradually add the dry ingredients to the wet ingredients, mixing on low speed just until a soft dough forms. Avoid overmixing.
Gather the dough into a ball, flatten it slightly into a disk, and wrap it tightly in plastic wrap. Refrigerate for at least 30 minutes, or up to an hour.
Preheat oven to 350 degrees Fahrenheit and line two baking sheets with parchment paper. Portion the chilled dough into 24 equal balls, about 1 1/2 tablespoons each, and place them a few inches apart on the prepared baking sheets.
Use your thumb or the back of a small spoon to press a deep well into the center of each dough ball, being careful not to press all the way through. Spoon about 1 teaspoon of the cooled apple filling into each indentation.
In a small shallow bowl, stir together the granulated sugar and cinnamon. Lightly roll each dough ball in the cinnamon sugar mixture, or sprinkle a pinch over the top of each filled cookie.
Bake the cookies for 11 to 13 minutes, until the edges are just turning golden and the centers look set. Let them rest on the pan for 5 minutes before moving them.
Transfer the cookies to a wire cooling rack and let them cool completely, about 15 to 20 minutes. Warm the caramel sauce slightly and drizzle it over the top of each cookie in a thin, even stream. Let the caramel set before serving or storing.