Preheat oven to 425°F.
Cook ground beef and onions in a large pan on medium-high heat for 10 to 15 minutes or until beef is fully cooked and onions are tender. Drain excess fat if needed. Add chili starter and tomatoes. Simmer uncovered for 3 to 5 minutes or until thick.
Combine 1 cup flour, undissolved yeast, sugar and salt in a large bowl. Add very warm water and oil; mix until well blended, about 1 minute. Gradually add enough remaining flour to make a soft dough. Dough should form a ball and will be slightly sticky. Knead** on a floured surface, adding additional flour if necessary, until smooth and elastic, about 4 minutes. (If not using Pizza yeast, cover dough and allow to rest for 10 minutes.)
Pat dough with floured hands to fill greased pizza pan or baking sheet. OR roll dough on a floured counter to 12-inch circle; place in greased pizza pan or baking sheet. Form a rim by pinching the edge of the dough.
Spread chili mixture over top of pizza crust. Sprinkle green chilies on top of chili mixture. Combine cheeses in a bowl and sprinkle over top of pizza. Garnish with additional onion and green bell pepper, if desired.
Bake on lowest oven rack for 25 to 35 minutes, until cheese is bubbly and crust is browned.
**To knead the dough, add just enough flour to the dough and your hands to keep the dough from sticking. Flatten dough and fold it toward you. Using the heels of your hands, push the dough away with a rolling motion. Rotate dough a quarter turn and repeat the “fold, push and turn” steps. Keep kneading dough until it is smooth and elastic. Use a little more flour if dough becomes too sticky, always working the flour into the ball of dough.