Chickpea Flatbreads with Burst Tomato Sauce

Name: Chickpea Flatbreads with Burst Tomato Sauce
Ingredients: 1 cup chickpea flour; 10 Tbsp. extra-virgin olive oil, divided, plus more for serving; 1½ tsp. kosher salt, divided; 2 pints cherry tomatoes, halved if large; 2 garlic cloves, finely chopped; 1 (15-oz.) can chickpeas, drained, rinsed; 1 Tbsp. dried oregano; 1 tsp. sherry or red wine vinegar; ¼ tsp. smoked paprika (optional); 2 Tbsp. cold unsalted butter, cut into pieces; 1 cup coarsely crumbled feta (about 5 oz.); Baby greens, such as arugula, kale, or spinach, and lemon wedges (for serving)