1poundassorted radicchiosuch as Verona and Treviso, heads halved lengthwise or quartered if large
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Heat a grill or grill pan until medium-hot. Meanwhile, bring a large saucepan of water to a boil. Add beets; boil until just tender, about 5 minutes. Drain; pat dry with paper towels.
Toss beets with 1 tablespoon oil in a medium bowl; season with salt, and set aside. Toss eggplants, zucchini, summer squash, bell peppers, and radicchio with remaining 1/4 cup plus 1 teaspoon oil in a large bowl; season with salt.
Grill all vegetables, turning once, until tender and golden brown, 4 to 8 minutes (cooking time will vary among vegetables). Arrange vegetables on a large platter. Serve warm or at room temperature; just before serving, spoon vinaigrette on top, and gently toss.