Combine 1 cup flour, sugar, milk powder, undissolved yeast, salt, black pepper, garlic powder, basil and oregano in a large bowl. Heat water and butter until very warm (120° to 130° F). Gradually add to flour mixture. Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Stir in onion, peppers and enough remaining flour to make a soft dough. Knead on a lightly floured surface until smooth and elastic, about 8 to 10 minutes. Cover, let rest 10 minutes.
Punch dough down; knead in pepperoni, bacon, mushrooms and cheese. Divide dough into 16 equal pieces; shape into balls. Place on greased baking sheets. Cover; let rise in warm, draft-free place until doubled in size, about 45 minutes.
Mix Egg Glaze ingredients with a fork in a small bowl: brush on rolls. Bake at 350°F for 20 to 25 minutes, or until done. Remove from sheets; serve warm.