Pour half the olive oil into a medium nonstick skillet on high heat and cook until it almost reaches the smoking point. While the oil is heating, cut the onion quarters in half crosswise, creating triangles. Cut off the ends and separate the layers with your hands. When the oil is ready, add the onions and bell peppers and cook on high heat for 2 minutes. Flip the pieces over and cook for 2 more minutes. Add garlic and basil and cook, stirring for 1 minute. Add 1 tablespoon of the vinegar, stir and cook for 30 seconds, remove from the heat and transfer to a bowl. In a large salad bowl, mix the greens with the remaining olive oil, remaining vinegar, the salt, pepper and feta cheese. Toss until everything is well coated. Divide the salad among the serving plates and top with the warm pepper-onion mixture.