In a large bowl, stir together yeast, 1 tablespoon all-purpose flour, and 1/4 cup warm water and let stand until surface appears creamy (5 minutes). Add whole wheat and remaining all-purpose flour, remaining 1/2 cup water, salt, and oil and stir until smooth. Add a bit more flour if necessary for dough to begin to pull away from side of bowl.
Knead dough on a floured surface until smooth, soft, and elastic (about 8 minutes). Form into a ball and place in a lightly oiled bowl. Cover with plastic wrap or a kitchen towel and let rise in a warm area until doubled (about 1 hour). Forty-five minutes before baking pizza, put a (12- x 17-inch) baking sheet on oven rack in lower third of oven and preheat to 500°.
Heat 1 tablespoon olive oil in a small nonstick skillet over medium-high heat. Sauté pepper strips with salt until just tender (6-8 minutes).
Once dough has risen, dust with flour (do not punch dough down), and divide into 4 equal pieces. Place one portion of dough in center of a piece of 8-inch-wide parchment paper. Pat dough evenly with your fingers and stretch into a 7-inch round, reflouring if necessary. Repeat with remaining 3 pieces of dough, each on its own piece of parchment. Place 2 pizza rounds on a large baking sheet. Top each with 1/4 of the tomato slices, onion, pepper strips, then cheese.
Carefully slide pizzas on parchment onto hot baking sheet in oven. Bake until dough is crisp and browned and cheese is golden (about 12 minutes). Transfer pizzas on parchment to cutting board (leave hot baking sheet in oven). Cool 5 minutes. Sprinkle with some basil. Make last 2 pizzas with remaining ingredients.