Over low heat, melt 1 Tablespoon of butter and 1 Tablespoon of olive oil in a large skillet. Add the onion and a pinch of salt and cook, stirring often, until onion starts to caramelize, about 30 minutes. Add a splash of balsamic vinegar and continue cooking for another 15 minutes. Remove the onions to a plate and set aside.
Preheat the oven to 400 degrees.
Add the remaining butter to the skillet, then saute the peaches and thyme over medium heat for roughly 8 minutes, until the peaches get a bit golden along the edges. Remove to a plate and set aside.
In the same skillet, heat the remaining olive oil over medium heat and then add the raw shrimp. Season with salt and pepper and saute until the shrimp is cooked through, about 5 minutes. Remove to a cutting board and chop roughly.
Arrange four tortillas on a baking pain lined with tin foil. Evenly distribute the peaches, shrimp, onions and brie between the tortillas. Top with the remaining four tortillas and cover tightly with another sheet of foil. Bake for 15 or until the cheese is gooey and melted. Remove the top piece of tin foil and let quesadillas sit for 5 minutes before cutting into quarters and serving.