For the dipping sauce: Combine the mayonnaise, horseradish, BBQ seasoning, mustard and some salt and pepper in a small mixing bowl. Store refrigerated in an airtight container until ready to serve.
For the onion blossom: Preheat a grill to 350 degrees F. Spay a disposable pie plate with olive oil cooking spray.
Peel the onion and trim the root end to remove any hairs. Slice the pointy stem end of the onion off. Turn the onion so the root end is facing up. Leaving a 1/2-inch rim around the root, make 4 cuts into the onion, down to the stem end. This results in 4 sections of the onion. Repeating the same motion, make 3 more cuts into each section of the onion, from root to stem end. Turn the onion over to reveal 16 onion petals. Use your hands to separate the layers of petals as needed.
Place the onion stem-side up in the prepared pie plate. Generously spray all of the petals with the olive oil cooking spray. Sprinkle the onion with the BBQ seasoning.
Put the onion on the grill and close the grill. Roast at 350 degrees F, checking halfway, until the onion blossom has opened, 40 to 45 minutes. The petals should be tender and darkened on the ends. Carefully remove the onion from the grill. Serve warm with the dipping sauce.