Cook the wheat berries: Bring a medium saucepan of water to a boil. Add the wheat berries, and cook until plump and tender, 35 to 40 minutes. Drain the wheat berries, and transfer them to a medium bowl. Toss with the sesame oil and 1/2 teaspoon salt. Serve warm or at room temperature. (The wheat berries can be cooked and refrigerated in an airtight container up to 2 days ahead. Microwave just enough to take the chill off, about 2 minutes, stirring about halfway through.)
Make the miso sauce: While the wheat berries cook, whisk together the mayonnaise, miso, mirin, vinegar and 1 tablespoon water in a small bowl until smooth.
Toast the sesame seeds: Swirl the sesame seeds in a medium nonstick skillet over medium heat until toasted and golden, about 3 minutes. Pour them into a small bowl.
Cook the salmon: Season the salmon with salt. Add the oil to the skillet, set over medium-high heat and heat the oil until it just starts to smoke faintly. Arrange the salmon, skin-side down, in the skillet. As the salmon cooks you'll see the dark orange flesh turn a paler orange. Once the lighter color reaches about 2/3 up the side of each piece, turn the salmon over; continue to cook for about 1 minute more (for medium-rare) for thicker pieces, 30 seconds more for thinner pieces. As each piece finishes, remove it from the skillet and transfer it to a plate.
Build the bowls: Evenly divide the wheat berries among 4 bowls. Make neat piles and/or rows of the cabbage, avocado and scallions. Top each with a piece of salmon and some miso sauce, a sprinkling of sesame and some seaweed-snack strips. Serve each bowl with a lemon wedge.
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