The day before: Start making the dough one day in advance; it needs to be refrigerated overnight. It can be made 2 days ahead. Keep chilled.
To make the dough:
Pour 2 tablespoons warm water into large bowl of stand mixer fitted with dough hook; sprinkle yeast over. Let stand until yeast dissolves, about 15 minutes (mixture will not be foamy).
Add both flours, 1/2 cup cool water, and 1 teaspoon coarse salt; mix on medium-low speed 4 minutes. Let rest 5 minutes, then mix on medium speed until dough is smooth, elastic, and slightly sticky, about 3 minutes.
Lightly oil medium bowl. Gather dough into ball and transfer to prepared bowl; turn to coat with oil. Cover bowl with plastic wrap and let rest at room temperature 30 minutes. Chill dough overnight.
Transfer bowl to warm, draft-free area and let dough rise, covered, until very slightly puffy, at least 2 hours.
To make the topping:
Place pizza stone or rimless baking sheet in oven; preheat to 500°F.
Sprinkle pizza paddle or another baking sheet generously with cornmeal. Roll out dough on lightly floured surface to 12-inch round; transfer to paddle.
Brush dough with oil; scatter peppers, then olives over. Sprinkle with Parmesan and rosemary. Arrange onion rings atop pizza, spacing apart.
Slide pizza onto stone or baking sheet in oven. Bake until lightly browned but not crisp, about 7 minutes. Remove pizza from oven and gently crack 1 egg into each onion ring.
Return pizza to oven and continue to bake until eggs are softly set and crust is golden, about 6 minutes. Sprinkle pizza with salt and pepper. Scatter arugula over top.