Set the oven to 350°F and allow it to heat while you prepare the onion. A fully heated oven gives the covered baking stage a consistent start.
Remove the papery outer layer and trim about a half inch from the stem end, leaving the root base untouched. The exposed top should reveal clean onion layers without disturbing their attachment at the bottom.
With the root base down, divide the onion from the top into eight equal sections, ending each cut about an inch above the base. The uncut lower portion should remain solid enough to anchor every section.
Ease the sections outward with your fingers to open the blossom. The petals should separate enough for coating while staying connected at the bottom.
Use a fork to blend the coconut flour with the paprika, garlic powder, cayenne, salt, and pepper in a mixing bowl. The mixture should have an even color with no concentrated pockets of spice.
Beat the eggs in a separate bowl, roll the onion through them, and spoon egg into the gaps between sections. Both the outer petals and the recessed inner surfaces should look wet.
Roll the onion gently in the seasoned flour, transfer it to a prepared baking sheet, and sprinkle the remaining mixture into uncovered areas. A light, even coating should reach the inner layers as well as the outside.
Apply the olive oil with a brush over the top and outer petals using light strokes. The exposed coating should look lightly moistened rather than scraped or streaked.
Enclose the onion in foil for a 30-minute bake, then remove the foil and continue baking uncovered for 10 minutes. The petals should soften during the covered stage and begin drying and coloring once exposed.
For extra crispness, finish under the broiler for 3–5 minutes while watching continuously. Stop when the exposed edges turn golden to rust-gold, before the fine tips blacken.