Prepare carmelized onions: dice onions in small pieces; melt 1 T. unsalted butter and 1 T olive oil in frying pan; on low heat, add 1/2 onions, sprinkle with 1/2 t. salt, add the rest of the onions and another 1/2 t. salt. Cover and leave onions undisturbed for 15 minutes. When onions are translucent, uncover and cook on medium low for 30 to 45 minutes, stirring occasionally until onions are golden brown.
Preheat oven to 425.
Brush flatbread with olive oil, spread carmelized onions on top and put small spoonsful of goat cheese on top of onions. drizzle with balsamic reduction.
Bake for about 8 minutes, cut in pieces and serve.