1. Combine the first 6 ingredients in a small saucepan; bring to a boil. Remove from heat; add cucumber. Let stand 30 minutes. Drain.
2. Combine panko and the next 7 ingredients (through egg white) in a bowl, and stir well. Divide the mixture into 4 equal portions, gently shaping each into a 1/2-inch-thick patty. Press a nickel-sized indentation in the center of each patty.
3. Heat a large cast-iron skillet over medium-high heat. Add sesame oil to pan; swirl to coat. Add patties; cook patties for 3 minutes on each side or until desired degree of doneness. Brush tops of patties evenly with hoisin; cook for 30 seconds.
4. Place 1 patty on bottom half of each bun; top each patty with 6 cucumber slices and top half of bun.