Preheat the oven to 350 degrees F (175 degrees C). Line 3 to 4 baking sheets with parchment paper.

Beat butter and sugar together in a bowl using an electric mixer until pale and fluffy. Beat in vanilla extract. Reduce mixer speed to low and gradually beat in flour and salt. Press dough together in plastic wrap; roll dough into 1 1/2-inch balls.

Place beaten eggs in one small dish and pecans in another dish. Dip each dough ball into the eggs and roll in pecans. Place dredged balls on the prepared baking sheets; press your thumb into the center of each ball.

Bake in the preheated oven for 10 minutes. Press each cookie again, with a spoon. Return to the oven and continue baking until golden, about 10 minutes more. Cool on wire racks.

Combine caramels and heavy cream in a small saucepan over low heat. Cook, stirring constantly, until caramels are melted and mixture is smooth, 4 to 6 minutes. Spoon the mixture into the thumbprints of the cookies and sprinkle with sea salt.