Beat butter and sugar with an electric mixer until fluffy. Mix in egg and vanilla extract. Add flour and salt and mix until just incorporated; overmixing will make the dough too stiff. Cover the dough with plastic wrap. Refrigerate for 30 to 60 minutes, or up to overnight.

Preheat the oven to 375 degrees F (190 degrees C).

Press about 1 teaspoonful of the chilled dough into the bottom and up the sides of each walnut-shaped mold. Place the molds on a baking sheet.

Bake in the preheated oven until browned around the edges, 12 to 18 minutes. Cool for a few minutes. Lightly tap on the bottom of the mold to pop out the cookies. Let cool until easily handled.

Spoon some dulce de leche onto the bottom of each cookie and sandwich the halves together.