Mix 2 cups flour, undissolved yeast, sugar and salt in a large mixer bowl. Add very warm water and butter to dry ingredients; beat 2 minutes at medium speed with electric mixer, scraping bowl occasionally. Add 1 cup flour, or enough flour to make a thick batter. Beat at high speed 2 minutes, scraping bowl occasionally. Stir in cheese, onion and enough additional flour to make a soft dough. Turn out onto lightly floured surface; knead until smooth and elastic, about 8 to 10 minutes. Cover; let rest 10 minutes.
Divide dough into 20 equal pieces. Form each piece into a smooth ball and place on greased baking sheets. Flatten balls to 3 to 4-inch rounds; cover. Let rise in warm, draft-free place until doubled in size, about 45 minutes.
Bake at 400°F 15 to 20 minutes or until done. Remove from baking sheets and cool on wire racks.
Optional Toppings: Whisk 1 egg with 1 tablespoon water. Brush on rolls just before they are ready to go into the oven. Sprinkle with 1/2 cup shredded sharp Cheddar cheese.
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