To make caramelized onions: In pan over medium-high heat, heat butter and olive oil. Add onions and let sit about 5 minutes. Once onions start sweating, turn heat to low and cover pan. Cook onions to deep golden brown, stirring every 10-15 minutes. With 10 minutes left, stir in balsamic vinegar.
To make flatbread: Preheat oven to 425 F.
On flatbread, layer cheese, bacon and caramelized onions. Bake 13-15 minutes, or until cheese is bubbly and edges are golden brown.
Toss arugula with olive oil and lemon juice, if desired, before placing on flatbread.