1. First, prepare the bean purée. Pick over and rinse the beans and put them in a bowl with the piece of kombu, add enough water to cover by about 2 inches, and set aside to soak for 6 to 8 hours or overnight.
2. Drain the beans, reserving the kombu, and transfer them both to a soup pot. Add 6 cups of water and bring to a boil over high heat. Reduce the heat and simmer the beans, uncovered, for 40 to 50 minutes, until just tender. Drain and set aside, reserving about a cup of cooking liquid.
3. Transfer the beans to a food processor or blender. Add the garlic, chili powder, sea salt, lime juice, cilantro, and a bit of the cooking liquid and pulse until smooth. Taste and adjust the seasonings. The purée canbe made up to 3 days ahead of time and kept in the refrigerator.
4. Preheat the oven to 350°F.
5. Prepare the quesadilla filling: Toss the mushrooms and pepper in olive oil and salt and pepper to taste. Place on a baking sheet in a single layer and roast for 30 to 45 minutes. Set aside.
6. Assemble the quesadillas: Spread a thin layer of the bean purée onto a tortilla. Top with half the roasted mushroom-pepper mixture and half the onion slices. Sprinkle half the cheese over the vegetables and repeat with the second tortilla. Fold the tortillas in half.
7. Bake the quesadillas until warmed through, about 15 minutes, turning once. Or, for a crisper result, broil or grill the quesadillas for about 5 minutes per side. Cut into quarters and serve at once.