Heat grill to medium-high.
Drain and rinse beans. Place into a food processor along with the Worcestershire sauce, onion, and egg white; process until smooth.
Add bread crumbs and pulse until well-combined.
Place sesame seeds in a shallow dish. With wet hands, shape bean mixture into 4-6 balls (depending on desired thickness of burgers), roll in sesame seeds to coat, then form into patties.
Place patties on grill and cover. Grill 6-8 minutes per side until patties are lightly browned and seeds are toasted.
Mash avocado and shred carrots. Toast buns if desired.
Add some avocado, carrots, a bean patty, and barbecue sauce to each bun, forming sandwiches.
Serve bean burgers with strawberries on the side.