Combine the first 6 ingredients in a small saucepan; bring to a boil.
Remove from heat; add cucumber.
Let stand 30 minutes.
Drain.
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.
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.
Place 1 patty on bottom half of each bun; top each patty with 6 cucumber slices and top half of bun.