Slice tops off onions. Leave root end in tact. Slice onions vertically, from the top to almost to the bottom, stopping the cut at about 1/2 inch before the root. Repeat cuts until there is a blossom-like onion. Carefully separate petals and place on a flat grill pan or flat foil baking sheet.
Preheat grill over medium heat. Combine paprika, garlic, cayenne, italian, and asiago until well combined. Sprinkle over onions. Drizzle onions with oil. Grill 15-30 minutes or until onions are soft and beginning to char. Remove from heat and sprinkle with parsley.