Heat 1 tablespoon oil in large skillet over medium-high heat. Add mushrooms and sauté 3 to 5 minutes or until tender. Stir in spinach, sauté an additional 30 seconds; remove from heat.
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 using RapidRise Yeast, cover with a towel and let dough rest for 10 minutes before proceeding to next step.)
Working from center to edges, press dough into and 1-inch up sides of a 12 to 14-inch greased deep-dish pizza pan; set aside.
Combine eggs, cream, basil, salt and pepper in a mixing bowl. Stir in sauteéd vegetables, roasted red bell peppers and Swiss cheese. Pour onto pizza crust; top with green onions.
Bake on lowest oven rack for 20 to 25 minutes, until eggs are set and crust is browned.
*If you don't have a thermometer, water should feel very warm to the touch.
**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.