Slice each hard-boiled egg in half lengthwise. Scoop the egg yolks into a medium bowl and mash with a fork. Add the mayonnaise, mustard, adobo sauce and paprika and stir together until completely combined; season with salt and pepper. Transfer the mixture to a piping bag and refrigerate until ready to use.
Lightly brush the egg whites on the cut sides with mayonnaise. Put the eggs cut-side down in a grill basket; close and secure the basket. Place the basket directly over the hot coals and cook until grill marks appear on the egg whites, 30 seconds to a minute. Transfer the whites to a serving plate and refrigerate until ready to serve.
When ready to serve, pipe some of the yolk mixture into each egg white and dust with a little paprika.
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