Combine eggs, syrup, melted butter, and vanilla in a large mixing bowl. Combine sugar and flour; add to egg mixture, mixing well. Set aside.
Roll pastry into a rectangle 1/4- inch thick on a lightly floured surface. Spread pastry with softened butter, leaving a 1-inch margin at edges. Fold in thirds like a letter; pinch edges to seal. Turn sealed edges to bottom.
Roll again to 1/8-inch thickness on a lightly floured surface. (If butter breaks through dough, flour lightly and continue rolling.) Cut into 4-inch rounds; fit rounds into 2 3/4-inch muffin tins. Press flat to cover sides of tins; trim pastry along edge of tin with a sharp knife. Prick bottom of each shell.
Sprinkle each unbaked tart shell with 1 teaspoon pecans. Spoon 2 tablespoons reserved filling into each tart shell. Bake at 350° for 35 minutes. Cool completely before removing from pans.