For the marinade: In a nonreactive bowl, combine the olive oil, garlic, cayenne, mustard, and basil. Rinse the chicken breasts, pat them dry, and add them to the marinade; turn to coat. Cover and refrigerate for 1 to 4 hours.
For the coating: In a large, shallow dish, mix the chips, crackers, basil, and pepper. Drizzle the melted butter over the crumb mixture and toss until well combined.
To coat and cook the chicken: Heat the oven to 450°F and butter a baking sheet or rack. Take a breast from the marinade with one hand--this is now your "wet" hand. Don't wipe off the marinade. Lay the chicken on the crumbs. Scoop and pat the crumbs over the breast using your other hand (your "dry" hand), patting until both sides are thoroughly coated. Put the breast on a buttered baking sheet or rack and repeat with the remaining breasts. Roast the chicken until it's crisp, browned, and cooked through, 25 to 30 minutes. Check after 15 minutes. If the chicken is getting too brown, reduce the heat to 400° and add 5 minutes to the total cooking time.