1. Make the shrimp: Bring a large pot of salted water to boil over high heat. Add shrimp, cook for 60 seconds, and drain. In a large, nonreactive container, layer shrimp and aromatics. Pour liquid ingredients over top while shrimp are still warm. Cover and refrigerate at least 3 hours to allow shrimp to pickle.
2. Make the eggs: Bring a large pot of water to boil over high heat and carefully lower eggs into boiling water. Set a timer for 8 minutes, and while eggs are cooking, prepare an ice bath in a large bowl. After 8 minutes, drain and shock eggs in ice bath, then peel them. Halve eggs and place yolks in the bowl of a food processor with mayonnaise, mustard, and lemon juice. Run food processor until mixture is completely smooth, about 2 minutes, and then adjust seasoning.
3. To assemble: Spoon filling into a zip-top bag, snip end from bag, and pipe filling into egg halves. Drain pickled shrimp and vegetables, chop pickled vegetables finely, and top each egg half with a pinch of chopped vegetables and a pickled shrimp.