Combine strawberries and sugar in a food processor and process until smooth.
Combine lemon juice and cornstarch in a small bowl, whisking until completely smooth. Add to the food processor with the strawberry mixture and process until smooth.
Strain strawberry mixture through a fine mesh sieve into a medium saucepan, discarding the seeds.
Bring strawberry mixture to a simmer over medium-low heat. Stir frequently and cook until mixture begins to thicken, about 3 to 4 minutes.
Add butter a few pieces at a time, stirring until melted. When the strawberry curd is smooth and thickened, and coats the back of a spoon, transfer to a glass bowl. Cover with plastic wrap, placing it directly on the surface of the curd, and chill for at least one hour.
To make the cookies:
Cook butter in a heavy saucepan over medium heat, stirring constantly until it foams, turns clear, and then turns a deep brown, about 6 minutes.
Pour browned butter into a large mixer bowl and let cool for 10 minutes.
Whisk together flour, salt, and baking powder in a medium bowl.
Add sugar and vanilla bean paste to mixer bowl with brown butter. Beat at medium speed for 1 minute, until well combined. Beat in egg yolks.
Reduce speed to low and gradually add flour mixture just until incorporated.
Add milk and mix on low speed until dough comes together in large pieces.
Line two baking sheets with parchment paper and divide dough in half.
Working with one portion of dough at a time, roll the dough out between 2 pieces of lightly floured parchment paper to just shy of 1/4-inch thick.
Use a floured 2 1/4-inch round fluted cookie cutter to cut out rounds. Gather scraps, re-roll, and cut until you have 16 rounds. Repeat with remaining disk of dough to create 32 total rounds.
Use a floured 1-inch round cutter to remove centers from 16 rounds.
Use a metal spatula to place rounds on baking sheets and chill for 30 minutes. Preheat oven to 350°F.
Bake, rotating pans halfway through, for 11 to 12 minutes, until lightly golden and edges are set.
Cool cookies for 10 minutes on baking sheets, then transfer to a wire rack to cool completely.
Turn over whole cookies and pipe strawberry curd in the center of each cookie. Smooth the curd with an offset spatula.
Dust cutout cookies with powdered sugar and gently sandwich cookies together. Store leftover cookies in a single layer in an airtight container in the refrigerator.
Strawberry curd can be made ahead and chilled for 1-2 days before assembling cookies.