1. Preheat oven to 375°. Rinse squash and pat dry. Cut around stem to make a 4-inch lid. Remove lid and scrape out and discard seeds. Brush interior of squash and inside of lid with 1 tablespoon olive oil. Sprinkle lightly with salt and pepper. Set lid on squash and place in a 10- by 15-inch baking pan. Bake for 10 minutes.
2. Meanwhile, peel onions and cut lengthwise into 3/4-inch-wide wedges. Brush onions with remaining 2 tablespoons olive oil and sprinkle lightly with salt and pepper. Arrange onions around squash and return to oven. Bake until squash and onions are tender when pierced, 25 to 30 minutes for onions and 35 to 40 minutes for squash; remove when done.
3. Meanwhile, rinse green beans and trim off ends; cut into about 3-inch lengths. Rinse, stem, and seed bell pepper half; cut into 1/4-inch-wide strips about 2 inches long.
4. Put butter in a 1- to 2-quart pan over medium-low heat; when it begins to melt, add garlic and stir occasionally until butter is melted, about 1 minute. Whisk in mirin, soy sauce, Worcestershire, cumin, and cayenne. Cook, whisking often, to blend flavors, 1 to 2 minutes. Pour into a bowl or small pitcher.
5. Shortly before serving, in a 4- to 5-quart pan over high heat, bring about 2 quarts water to a boil. Add beans and red pepper; cook just until crisp-tender to bite, 4 to 5 minutes. Drain well and season to taste with salt and pepper.
6. Transfer squash bowl to a serving bowl or platter. Cut it--down to but not through the base--into 10 wedges for a flower-like presentation; for a bowl shape, cut the squash three-quarters of the way to the bottom. Mound green bean-red pepper mixture and roasted onions in the center. Drizzle 1/4 cup of the mirin-soy mixture over the vegetables. Reserve the squash lid for another use or for a decorative top for the bowl. Offer remaining sauce to add to taste.
If you want to get a head start on this dish, cut and prepare the vegetables and squash up to 1 day ahead; cover and chill. Cook vegetables shortly before serving.