Preheat the oven to 350 degrees F (175 degrees C). Line baking sheets with parchment paper.

Whisk cornstarch, flour, baking powder, and baking soda together; set aside.

Beat butter and sugar together in a large bowl with an electric mixer until light and fluffy. Beat in 2 egg yolks, one at a time, beating well after each addition; beat in third yolk along with lemon zest, vanilla rum, vanilla extract, and lemon extract. Gently fold in cornstarch mixture with a spoon, making a crumbly dough, until cohesive enough; press into a ball. Wrap dough in plastic wrap; chill 30 to 60 minutes.

Roll dough out, using as little flour as possible, until about 1/4-inch thick. (The dough will have an unusual consistency.) Cut dough using a small round cookie cutter. Continue pressing dough together, rolling, and cutting until all dough is used. Place cookies spaced 1/2-inch apart onto the prepared cookie sheets.

Bake in the preheated oven until set but not browned, 7 to 10 minutes. Transfer cookies immediately to a wire rack to cool.

Spread 1 teaspoon dulce de leche onto underside of 1 cooled cookie; sandwich with second cookie until caramel oozes out sides. Roll sides in shredded coconut. Repeat with remaining cookies, dulce de leche, and coconut.