1. Preheat oven to 425°F. Unroll refrigerated Classic Pizza dough with parchment paper on baking sheet. Cut dough into 8 equal-sized triangles.
2. To make homemade Caponata: peel red onion and garlic clove and chop finely. In a saucepan, briefly braise onion and garlic in olive oil. Add finely diced eggplant (rinsed and dried) and finely sliced celery (cleaned and rinsed) and braise for another 3-4 minutes while stirring. Stir in coarsely diced tomatoes, vegetable stock, capers, olives, balsamic vinegar, and freshly squeezed lemon juice. Season with salt, cayenne pepper, and 1-2 pinches sugar and simmer over low heat for 6-8 minutes.
3. Distribute Caponata evenly over the triangles, leaving a small margin around the edges. Sprinkle each triangle with 1 Tbsp. grated Pecorino.
4. Place baking sheet on bottom rack and bake about 8 minutes until golden-brown, serve hot.