Chickpea Flatbreads with Burst Tomato Sauce - Fresh-Pressed Olive Oil

Ingredients

• 1 cup chickpea flour
• 10 tablespoons extra virgin olive oil, divided use, plus more for serving
• 1 1/2 teaspoons kosher salt, divided
• 2 pints cherry tomatoes, halved if large
• 2 garlic cloves, finely chopped
• One 15-ounce can chickpeas, drained (see Note), rinsed
• 1 tablespoon dried oregano
• 1 teaspoon sherry or red wine vinegar
• 1/4 teaspoon merquén or smoked paprika (optional)
• 2 tablespoons cold unsalted butter, cut into pieces
• 1 cup coarsely grated queso fresco or crumbled feta (about 5 oz.)
• Baby greens, such as arugula, kale, or spinach
• Lemon wedges, for serving