Make the cookie dough. In a large bowl, cream together softened butter, powdered sugar, and brown sugar until light and fluffy. Add egg, vanilla extract, and almond extract, mix well. Add sourdough starter and mix until combined. In a separate bowl, whisk together flour, baking powder, salt, and cream of tartar. Gradually add dry ingredients to wet ingredients, mixing until just combined. Wrap in plastic and refrigerate for 12 to 48 hours.

Make the fermented crumble. In a medium bowl, mix flour, brown sugar, salt, powdered sugar, and sourdough starter. Add cold cubed butter and cut in using a pastry cutter or your fingers until mixture resembles coarse crumbs. Add vanilla and mix gently.

Prepare the strawberry compote. In a medium saucepan, combine sliced strawberries, granulated sugar, cornstarch, and lemon juice. Cook over medium heat, stirring gently, for 5 to 7 minutes until mixture thickens and becomes glossy. Strawberries should hold their shape but be coated in thick, jam-like syrup. Remove from heat and let cool completely, at least 30 minutes.

Preheat oven to 350°F (175°C). Line a large baking sheet with parchment paper.

Assemble the cookies. Using your circle cutter as a guide, spread about 1 heaping tablespoon of fermented crumble mixture on the parchment in circles. Top each crumble circle with 2 teaspoons of cooled strawberry compote. Roll out the fermented cookie dough on a floured surface to a rough rectangle, approximately ¼ to ½ inch thick. Using your circle cutter, cut out circles from the dough. Place each cookie circle on top of the crumble and strawberry layers. Gently press down to adhere.

Bake for 18 to 22 minutes, until cookie tops are golden brown and set. Remove from oven and let cool completely on the baking sheet, at least 1 hour.

Melt white chocolate with 1 to 2 teaspoons coconut oil in a microwave or double boiler until smooth. Drizzle melted white chocolate over the cooled cookies.

Serve immediately or store covered for up to 3 days.