Place a large skillet over medium to medium-high heat. In a bowl, toss together the red onion, shallot, and 2 tablespoons of olive oil until coated.
Dump the onion and oil into the hot skillet and saute until they just begin to soften and turn translucent, around 3 minutes.
In the same bowl, toss the tomatoes and 2 tablespoons of olive oil. Then add both to the skillet with the onions. Sprinkle with salt and pepper and allow to cook, stirring occasionally, until the tomato skins blister and soften, around 5 minutes.
Add the fresh basil, stir gently, and then remove from heat. Adjust the seasoning with more salt and pepper as needed. Sprinkle with feta cheese. Serve warm with your favorite meals.