Heat the oven to 400 degrees F. Thinly slice one shallot. In a large bowl, toss together the sliced shallot, cauliflower, broccoli, and half the Brussels sprouts with enough olive oil to lightly coat (about ⅓ cup). Season with salt and pepper. Divide between two baking sheets, spread out, and roast, stirring occasionally, until fully tender and charred in spots, 30 to 35 minutes. Reserve the bowl (no need to clean it).
While the vegetables roast, make the dressing: Finely grate the remaining shallot into the reserved bowl to produce two teaspoons. Add ginger, rice vinegar, soy sauce, sesame oil, and ¾ cup olive oil. Season with salt and pepper to taste.
Thinly slice the remaining Brussels sprout halves and transfer to the large bowl along with the kale. Toss with the dressing, massaging the kale to wilt slightly.
Once the vegetables have finished roasting, add to the large bowl and toss to coat in the dressing. Taste and adjust seasonings as needed.