Whisk together all of the vinaigrette ingredients and let stand for 1 hour before using. Shake well just before dressing the dish.
Place the cucumbers on a parchment-lined baking sheet and sprinkle with salt. Let them sit for 30 minutes, then pat dry with paper towels.
Rinse the rice in the sink until the water runs clear. Add the rice and water to a saucepan, bring to a simmer, cover, and cook until the grains are tender, about 30 minutes.
Preheat a grill to medium heat. Season the salmon with salt and pepper and grill, skin-side down for 8 minutes, flip, then continue cooking until cooked through, 4-5 minutes more.
Whisk together the sugar and rice wine vinegar for the cucumbers in a bowl until the sugar dissolves. Add the cucumbers.
In a serving bowl, combine the cooked rice, carrot matchsticks, green onions, edamame and vinaigrette. Toss to combine.
Add the butter lettuce leaves, pickled cucumbers, salmon and sesame seeds to the brown rice base. eak