Make the dough. Toward the end of the rising time, preheat the oven to 500˚ F (use the convection setting, if available) with a pizza stone, baking steel or inverted large baking sheet in the lower third of the oven.
Line a rimmed baking sheet with foil; put the bell pepper halves on one side and the mushrooms on the other side. Drizzle with 2 tablespoons olive oil; rub all over the peppers and toss the mushrooms to coat. Arrange the peppers cut-side down and roast, turning the peppers and stirring the mushrooms occasionally, until the peppers are tender and browned and the mushrooms are tender and dry, 15 to 20 minutes; season with salt and pepper.
Meanwhile, heat the remaining 2 tablespoons olive oil in a medium nonstick skillet over medium heat. Add the onions, 1 teaspoon salt and a few grinds of pepper. Cook, stirring, until tender and lightly browned, 15 to 20 minutes (reduce the heat if the onions start to get too dark). Stir in 1/4 cup water, the tomato paste and sugar. Cook, stirring frequently, until the tomato paste is toasted and a few shades darker, about 10 minutes. Remove from the heat and stir in the vinegar; season with salt, pepper and more sugar, if needed. (The sauce should be slightly sweet and tangy.) Let cool.
Remove the roasted peppers to a cutting board and chop; transfer to a large bowl and stir in the mushrooms, artichoke hearts and a pinch each of salt and pepper.
On a lightly floured surface, cut one 1-pound ball of dough in half. Cup your hand around each piece and gently roll each into a small ball. Transfer 1 ball of dough to a lightly floured piece of parchment paper and roll into a thin 12-inch round with a rolling pin (keep the other ball of dough covered).
Top the dough with half of the sauce, spreading it to the edge, then top with half of the roasted vegetable mixture and 2 tablespoons Parmesan.
Transfer the pizza (on the parchment) to the hot stone and bake until the crust is set, 3 to 5 minutes. Slide out the parchment from under the pizza. Continue baking until the crust is golden brown and crisp, 7 to 10 more minutes. Remove the pizza to a cutting board and dot with half of the ricotta. Drizzle with olive oil, season with salt and top with basil. Repeat to make another pizza.
All-Purpose Pizza Dough:
Sprinkle the yeast over the hot water in a small bowl. Let stand until dissolved and slightly foamy, 5 to 10 minutes.
Meanwhile, whisk the bread flour, all-purpose flour, sugar and salt in a large bowl and make a well in the center. Pour the yeast mixture into the well and add the olive oil. Stir with a wooden spoon until a shaggy dough forms.
Turn the dough out onto a lightly floured surface, dust the dough with flour and knead, dusting with more flour as needed, until very smooth and elastic but still slightly tacky, 3 to 5 minutes.
Transfer the dough to an oiled bowl, turning to coat. Cover with plastic wrap and let rise in a warm place until doubled in size, about 1 1/2 hours. Divide the dough into two 1-pound balls. (If making sheet-pan pizza, let the dough rise, covered, on an 11-by-17-inch rimmed baking sheet coated with 3 tablespoons olive oil instead of in a bowl; do not divide into two balls.)
If not using the dough right away, wrap in plastic and refrigerate for up to 2 days or freeze for up to 3 months. Bring to room temperature before using.