Brush bell peppers with the 1 tablespoon olive oil and sprinkle with thyme.
Place on rack of a broiler pan; broil about 4 inches below heat, turning often, until charred about 10 minutes.
Enclose peppers in a plastic bag and let stand for 15 minutes; then pull off and discard skins, stems, and seeds.
Cut peppers into quarters.
To prepare aioli, whirl peppers, garlic, salt, vinegar, and egg in a blender or food processor until smooth.
With motor running, add salad oil and the 1/2 cup olive oil in a thin, steady stream; continue to whirl until aioli is smooth.
If made ahead, cover and refrigerate for up to 1 week.
Rinse crab pieces, pat dry, and pile onto a platter.
Provide nutcrackers and small forks for removing meat from shells; dip shelled meat into aioli to eat.