Turn on the oven broiler to high with a rack in the center position. Line a rimmed sheet pan with foil.
Arrange the peppers, cut-side down, on the prepared sheet pan. Brush peppers with 1 tablespoon of the olive oil. Broil, rotating the pan at least once, until the tops of the pepper are completely charred, 8 to 10 minutes. Remove from the oven and let cool.
Once cooled, peel away the skins from the bell peppers and discard. Slice the peppers into thin strips and arrange on a plate.
Top peppers with prosciutto, torn burrata, remaining olive oil, parsley, salt and pepper. Serve cold along with crackers, crusty bread or as is.
