Heat broiler, with rack set 4 inches from heat. Line a rimmed baking sheet with aluminum foil. In a medium bowl, combine buttermilk, lemon juice, garlic, and Parmesan. Season with salt and pepper.
Place chicken in a resealable plastic bag; reserving 1/2 cup for salad, add buttermilk mixture. Refrigerate chicken at least 15 minutes and up to 1 day.
Meanwhile, make croutons: Place bread on prepared baking sheet. Brush both sides with oil, and season with salt and pepper. Broil until toasted, 1 to 2 minutes per side; cut into 1-inch pieces (and reserve baking sheet).
Transfer chicken to baking sheet; discard marinade. Broil until opaque throughout, 14 to 16 minutes. Let rest 5 minutes; thinly slice crosswise. In a large bowl, stir together mayonnaise and reserved buttermilk mixture. Add romaine, radicchio, chicken, and croutons; toss to combine. Serve immediately.
Bathed in Buttermilk: Low-fat buttermilk gives the chicken zip and tenderizes the meat. The longer the soak, the softer the chicken.