2 (8-ounce) balls burrata or fresh mozzarella, drained
Basil leaves (for serving)
1 teaspoon toasted sesame seeds
Flaky sea salt
Directions
Cook fava beans in a medium saucepan of boiling salted water until tender, about 4 minutes. Using a mesh sieve, transfer to a colander set in a bowl of ice water. Drain and peel. Combine lemon juice, 2 Tbsp. oil, and half of fava beans in a medium bowl and lightly mash with a fork. Stir in remaining fava beans; season with kosher salt and pepper.
Combine cucumbers and vinegar in a medium bowl; season with kosher salt. Let sit until slightly softened, 10–12 minutes.
Prepare a grill for medium-high, indirect heat (for a charcoal grill, bank coals on one side of grill; for a gas grill, leave one or two burners off); lightly oil grate. Working with 1 piece of dough at a time, gently stretch to about a 10x8" oval. Grill over direct heat, turning and rotating as needed, until bread is stiff and both sides are lightly charred, about 3 minutes total. Move to indirect heat to keep warm while you grill the remaining piece of dough.
Transfer flatbreads to a work surface. Tear burrata into pieces and divide between flatbreads; top with fava bean mixture, cucumbers, and basil. Sprinkle with sesame seeds, sea salt, and pepper.
Fava beans can be cooked 3 days ahead. Cover and chill.
