Whisk together the shallots, vegetable oil, vinegar, ginger, miso, sesame oil and sugar in a small bowl; set the marinade aside.
Halve the avocados lengthwise, remove the pits and score the interior of each half in a 1/2-inch crosshatch pattern. Set the halves, cut-side up, on a rimmed baking sheet, and sprinkle with some salt and pepper. Spoon the marinade into the avocado cups, dividing evenly (about 1 tablespoon per avocado half), and let stand for 5 minutes while it soaks in. Spread the marinade (with a pastry brush or the back of a spoon) over the entire cut surface of the avocados and gently into the crosshatch cuts.
Meanwhile, heat a grill or a grill pan to medium-high. Lightly oil the grill grates or grill pan. Grill the marinated avocado halves, cut-side down, for 2 minutes, gently loosen with a spatula, then continue grilling until the avocados show grill marks and the marinade is fragrant and caramelized, about 2 minutes more. Flip the avocados to their skin side, sprinkle with some salt and pepper and grill for 1 minute. Transfer to individual serving plates, and garnish each with some of the basil and a lemon wedge.
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