Even after they're grilled, the vegetables keep their clean, bright flavors. Arrange them in sections on several platters rather than mixing them together to create a geometric look.
Place the peppers directly over the flame of a gas-stove burner on high heat or on a grill. Roast peppers, turning with tongs, until blackened all over. (Alternatively, broil peppers on a baking sheet, turning as each side becomes charred.) Transfer to a large bowl, and cover immediately with plastic wrap. Set aside to steam 15 minutes. Peel peppers; discard skin. Remove stems, seeds, and ribs; cut each pepper into 2-inch pieces.
Prepare an ice-water bath; set aside. Bring a large pot of water to a boil. Add 1 tablespoon salt. Add carrots; cook until tender, about 8 minutes. Using a slotted spoon, transfer to ice-water bath to stop cooking; transfer to kitchen or paper towels to drain completely. Repeat the process with squash, zucchini, fennel, and celery root. Squash and zucchini will take about 5 minutes; fennel and celery root will take about 8 minutes.
Preheat grill to medium (if you are using a charcoal grill, coals are ready when you can hold your hand 5 inches above grill for just 5 to 6 seconds). Whisk together thyme, basil, garlic, and oil in a small bowl; season with salt and pepper. Brush all of the vegetables with herb oil. Grill peppers and blanched vegetables until heated through and marked by grill, about 5 minutes for peppers, 8 to 10 minutes for blanched vegetables. Grill radicchio and tomatoes until cooked through, 8 to 10 minutes for tomatoes, 7 to 8 minutes for radicchio. Transfer to serving platters. Brush with remaining herb oil; season with salt. Serve warm or at room temperature.
You can blanch vegetables a day ahead and refrigerate in airtight containers.