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Heat an outdoor grill to medium-high, direct heat (400 to 450ºF). Once hot, turn off one of the burners for indirect heat. (For charcoal grills, keep the hot coals on one side of the grill for indirect grilling.) Place both halves of the garlic head (from the dressing ingredients) in the middle of a piece of foil, drizzle with 1 tablespoon oil, and sprinkle with a big pinch of salt. Wrap the garlic in the foil and place on the grill over indirect heat. Cover and grill until a center clove of garlic is completely soft when pierced with a paring knife, 35 to 40 minutes.
Meanwhile, toss the broccoli and radishes with remaining 2 tablespoons oil, 1/2 teaspoon salt, and 1/2 teaspoon pepper in a large bowl. When the garlic is almost done, place the broccoli and radishes on the grill in a single layer over direct heat. Grill, turning occasionally, until charred in spots and crisp tender, 8 to 10 minutes.
Unwrap the garlic and squeeze the cloves into a large bowl; mash with a fork. Add the remaining dressing ingredients, season with salt and pepper, and whisk to combine. Transfer half the dressing to an airtight container.
Add the broccoli and radishes to the bowl with the remaining dressing and toss to coat. Arrange on a platter and sprinkle with toasted sesame seeds or peanuts. Serve with more dressing, if desired. Save the remaining dressing for leftovers, or to use on other grilled veggies or chicken.