Preheat oven to 325 degrees F.
Melt butter in a large frying pan over medium heat. Add onions and cook until golden brown. Set aside to cool. Unroll pie dough 1 at a time. Use a 2 1/2-inch round cutter and cut as many rounds as possible (about 14 to 16), reserving scraps repeat with second pie dough. Gather together scrap dough and cut out as many rounds as needed to make 24 in total. Fit dough rounds into cups of mini muffin pan. Place in the oven and bake for 10 to 12 minutes. Remove and set aside. In a small bowl, whisk together the egg, half-and-half, golden onion soup mix, and fines herbs. Set aside. Divide onions among tartlet shells, about 1/2 teaspoon each. Stir the egg mixture and fill tartlets to cover the onions. Top each with 1/2 teaspoon of cheese. Bake for 45 to 50 minutes.