1. To make the dough, in a large bowl, sprinkle yeast over 1 cup warm (about 110°) water. Let stand until yeast is softened, about 5 minutes. Add 2 1/2 cups flour and the salt; beat with a mixer on low speed until incorporated, then on medium-high speed until dough is stretchy, about 3 minutes.
2. If using a dough hook, beat on high speed until dough is smooth and elastic, about 5 minutes. If dough is still sticky, beat in more flour, 1 tablespoon at a time.
If kneading by hand, scrape dough onto a lightly floured board. Knead until smooth, springy, and no longer sticky, about 10 minutes. Add flour as necessary to prevent sticking.
3. Divide dough in half and shape each half into a ball. Set balls on a lightly floured surface in a warm place and cover loosely with plastic wrap. Let rise until doubled in size, 45 minutes to 1 hour.
4. Meanwhile, make filling: In a 10- to 12-inch frying pan over medium-high heat, stir onion in 1 tablespoon olive oil often until limp and beginning to brown, 3 to 5 minutes. Add radicchio and stir often until limp and beginning to brown, 5 to 7 minutes. Stir in wine and cook, stirring occasionally, until evaporated, 10 to 12 minutes. Add salt and pepper to taste. Let cool at least 20 minutes.
5. Punch each ball of dough down and knead lightly to expel air. On a lightly floured surface, with a lightly floured rolling pin, roll each ball into a 10- to 11-inch round. Sprinkle half the mozzarella over half of each round, to within 1 inch of edge; top cheese equally with radicchio mixture. Fold the bare half of each dough round over filling, aligning edges; pinch edges to seal. Starting at fold, roll each pizza into a tight log; pinch log ends together to seal. Transfer logs, seam down, to a 12- by 15-inch baking sheet, shaping into equal-size half-rings and spacing at least 3 inches apart.
6. Bake in a 425° regular or convection oven until golden brown, 25 to 30 minutes.
7. Meanwhile, make salad: In a bowl, mix tomatoes, basil, garlic, and remaining 2 tablespoons olive oil; add salt and pepper to taste.
8. With two wide spatulas, transfer each half-ring to a rimmed board or platter, fitting ends together to form a complete ring. Using a slotted spoon, mound about 2 cups tomato-basil salad in the center of the ring; set bowl with the remaining salad alongside board or platter. Cut the pizza ring into 1 1/2- to 2-inch sections and serve with salad.
Notes: Sangiovese is a great match for this appetizer; try the Vandale 1999.