Char the Poblano Chiles: Roast the poblanos directly over a gas burner or under a broiler, turning frequently, until the skin is evenly blistered and blackened (about 8 minutes). Transfer them to a large bowl, cover with a plate, and let them steam for around 15 minutes. Once cooled, use a paper towel to rub off the skin. Remove the seeds and stems, then slice the peppers into thin strips.
Cook the Bean Mixture: Heat a 12-inch skillet over medium heat and add 2 tablespoons of olive oil. Sauté the sliced onion with ½ teaspoon of salt, stirring occasionally, until it turns golden and soft (8–10 minutes). Stir in the tomato paste, minced garlic, and chili powder, and cook until fragrant (about 1–2 minutes). Add the black beans, 1 cup of water, and the remaining ½ teaspoon of salt. Mash about half the beans with a potato masher or fork and continue to cook until the liquid thickens and reduces (roughly 7 minutes).
Prepare to Broil: Position an oven rack as far from the heat as possible, and preheat the broiler.
Assemble the Quesadillas: Lightly brush one side of half the tortillas with olive oil and arrange them, oiled-side down, on a baking sheet. Spread about ⅓ cup of the bean filling evenly over each tortilla. Top with the poblano strips, shredded smoked mozzarella, fresh cilantro leaves, and a squeeze of lime juice. Place another tortilla on top of each, brushing the tops with olive oil.
Broil and Serve: Broil the quesadillas until they turn golden and crispy, about 3 to 5 minutes. Slice into wedges and serve warm with lime wedges, avocado slices, or your favorite accompaniments.