Make the strawberry compote. Combine strawberries, sugar, corn starch and lemon juice in a small saucepot and toss to coat evenly. Place over medium heat and cook for 5-8 minutes while stirring gently quite frequently until it thickens. Once it starts to bubble, let it bubble for about 15 seconds then transfer to a clean bowl to cool completely. Chill in the fridge for 1 hour. TIP: This can be made up to 3 days ahead and stored in a jar in the fridge.
Make the cookie dough. Combine soft butter with both sugars and vanilla in a large bowl and beat with an electric mixer on medium-high speed for 1-2 minutes until pale and fluffy. Add the egg and mix until well incorporated. Combine flour, baking powder, baking soda and salt in a medium bowl and whisk to blend evenly. Add these dry ingredients to the butter/sugar mixture and fold it in until evenly combined. Cover the bowl and refrigerate for 30 minutes to make it easier to roll the dough into balls and also help make thicker cookies.
Preheat your oven to 350°F. Line 2 large baking sheets with parchment paper.
Make the shortbread crumbs while the dough is chilling. Combine flour and both sugars (and salt if you’re using unsalted butter) in a small bowl and blend well so there are no lumps of brown sugar. Add the cooled melted butter and toss it all together until it forms clumps. I like to use my fingers for this part. Spread the crumbs out on a parchment-lined baking tray and bake for 12-15 minutes tossing halfway through so that they brown evenly. They are ready when they are evenly golden brown. Remove from the oven and set aside to cool completely. Break up the pieces further as desired.
Bake the cookies. Scoop the dough into balls with a large cookie scoop, roll the dough balls in granulated sugar so they are evenly coated. Place the dough balls onto prepared baking sheets spacing them at least 2 inches apart with room to spread. Bake for 11-13 minutes until evenly golden and the edges are browned. The centers will puff up and still feel slightly soft in the middle. They will deflate slightly as they cool. Transfer baking tray to a wire rack to cool for 2 minutes, then press an indent into the centers with the back of a teaspoon measure. Let cool a few more minutes on the tray then transfer to a wire rack to cool completely.
Make the vanilla glaze. Combine powdered sugar, cream and vanilla in a small bowl and whisk to blend well until smooth and glossy. Adjust the consistency with some milk as needed. I prefer it on the thicker side so it holds up and looks nice and white over the strawberries.
Assemble cookies. Spoon about 2 teaspoons of strawberry compote into the indents on the cooled cookies, sprinkle some shortbread crumbs on top and then drizzle the glaze over top.
Enjoy these cookies immediately once assembled. Store cookies in an airtight container in the fridge for 3-4 days.