Preheat oven to 400 degrees F (204 C) and position a rack in the middle of the oven. If using a pizza stone, place the stone in the oven as it preheats.
Sauté onion in butter or olive oil over medium heat until soft, stirring frequently. Season with salt and pepper. Cover to keep in the moisture if they begin to look dry – add a splash or two of water if they’re getting too brown. Set aside once cooked.
Once oven is preheated, brush flatbread with olive oil and place on the pizza stone or a baking sheet and “pre-bake” to crisp for about 5 minutes.
In the meantime, spoon goat cheese into a bowl and add a splash of milk or water to thin so it’s spreadable. Whisk until smooth.
Remove flatbread from oven and spread with goat cheese and top with onions – I used all of the onions but use less and reserve the leftovers if you prefer.
Place back in the oven and bake for 10 minutes, or until the edges appear crisp and the onions and goat cheese have warmed through.