Combine cottage cheese, 1 tablespoon butter, and 1 beaten egg; stir well, and set aside.
Combine flour, water, and 1/2 teaspoon salt in a medium bowl; add 2 eggs, stirring well.
Turn dough out onto a floured surface, and divide into 8 equal portions; roll each portion into a ball. Roll each ball into a 4-inch square. Place 1 tablespoon cheese mixture in center of each square. Moisten edges with water; fold over to form a triangle. Press edges together with fork tines to seal.
Place turnovers, a few at a time, in boiling water with 1 tablespoon salt. Cook 12 minutes, stirring occasionally to prevent turnovers from sticking to bottom of pan. Drain on paper towels; serve with melted butter, if desired.