To make caramelized onions: Preheat oven to 400 F.
In large stockpot over medium-low heat, warm oil and melt butter. Add onions and cover; cook 20-25 minutes, stirring occasionally. Add thyme sprigs and season with salt, to taste; turn pan lid slightly ajar. Place pot in oven 1 hour, stirring occasionally.
To make flatbreads: Preheat oven to broil. On sheet pan, toast flatbreads under broiler 4 minutes, or until toasted, flipping halfway through.
Spread 1/4 cup of caramelized onions over each toasted flatbread. Refrigerate leftover onions.
Tear mozzarella and place over onions. Divide prunes among flatbreads and place flatbreads under broiler 4-6 minutes until cheese has melted and is beginning to brown.
Sprinkle flatbreads with fresh thyme and season with salt and pepper, to taste.