Whirl egg, 1/2 tsp. salt, garlic, and mustard in a food processor until smooth. Add oil, drop by drop at first and then in a slow stream once mixture has begun to emulsify. Whirl until mixture is thick, then add red pepper flakes and lemon juice and pulse to combine. Chill aioli, covered with plastic wrap.
Mix shrimp with parsley, thyme, and salt to taste. Add just enough aioli to bind the shrimp (5 to 6 tbsp.); save leftover aioli for another use (see Notes).
Cut bases off endive and separate leaves (save small ones for salad). Arrange leaves on a platter and top each with a tablespoon of shrimp mixture and a grind of pepper. Serve immediately.
Note: Nutritional analysis is per serving.