Roasted garlic is milder and sweeter than raw garlic. The peppers and garlic can be roasted ahead of time; refrigerate them in a covered container until ready to use.
Heat broiler. Place peppers and garlic on a foil-lined baking sheet; broil 4 inches from heat source, skin side up, until peppers are blistered and blackened in spots and garlic is soft, about 20 minutes. Let cool, about 5 minutes. Use paper towels to rub off pepper skin; squeeze out garlic cloves (discard skins).
Make sauce: In a blender, combine peppers, garlic, vinegar, 2 tablespoons oil, and 3 to 4 tablespoons water; season generously with salt and pepper. Blend until emulsified.
Heat grill to medium; oil grates. Rub both sides of chicken breasts with remaining tablespoon oil; season generously with salt and pepper. Grill chicken, turning once, until browned and cooked through, 5 to 8 minutes per side. Serve with sauce, and garnish with arugula, if desired.
Instead of roasting whole peppers, quarter them, and remove the ribs and seeds before placing under the broiler. This way, there's no need to turn them as they cook.