Combine Special Sauce ingredients in a mixing bowl; refrigerate until ready to use.
Combine 1 cup flour, dry yeast, onions, sugar and salt in a large mixer bowl and stir until blended. Combine water and butter in a microwave-safe bowl. Microwave on HIGH in 15 second increments until warm but not hot to the touch (120° to 130°F). Butter will not melt completely. Add to flour mixture.
Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Stir in just enough remaining flour so that the dough will form into a ball.
Knead on lightly floured surface until smooth and elastic and springs back when lightly pressed with 2 fingers, about 6 to 8 minutes. Cover with a towel and let rest for 10 minutes.
Cut into 24 equal pieces. Roll each piece into an approximate 4-inch circle using a rolling pin. Fold one half of a slice of American cheese in half and place in center of dough. Top with one meatball. Fold up dough and pinch seams to seal. Place seam side down and a large baking sheet lined with parchment paper or coated with nonstick cooking spray. Cover with towel; let rise in a warm place 25 to 30 minutes. Dough will not double in size.
Bake in preheated 375°F oven for 20 minutes or until golden brown. Serve with pickle slices and sauce.