Start the rice and rice water in a covered saucepan; simmer on low for 15 minutes after boiling, then rest for 5 minutes.
Heat the oil in a large skillet over medium-high heat. Brown the beef for 7 to 8 minutes, leaving it in small clumps and cooking it to 160°F.
Add garlic and stir for 30 seconds. Mix gochujang, soy sauce, vinegar, sugar and glaze water, then add to the skillet and simmer for 1 to 2 minutes until sticky.
Wash and dry the lettuce and cut the cucumber while the beef cooks.
Put small spoonfuls of rice and beef into the lettuce leaves, add cucumber sticks and divide the sesame seeds over the filling. Serve three wraps per person with any remaining rice alongside.