Heat a sauté pan to medium-high. (Or coat grill racks with oil and preheat to medium-high.)

In a food processor, combine mushrooms, onion, beans, two of the garlic cloves, egg, rosemary, salt, and pepper. Process until well combined but not completely puréed; mixture will be wet. Transfer to a medium bowl and, using hands, mix in bread crumbs until mixture holds its shape but is still moist. Form into six patties and brush each side with olive oil. Panfry or grill for 10 minutes, turning halfway through, until crispy and browned on the outside and heated through.

While burgers are cooking, make pesto: In a food processor, combine remaining garlic clove, watercress, pine nuts, and olive oil. Pulse until a paste forms. Season with sea salt and pepper.

During final 2–3 minutes of cooking, toast bread to warm. Remove from pan or grill, and spread with pesto and goat cheese. Add burgers and serve immediately.