In a small bowl, whisk the mayonnaise with the ketchup, relish and Worcestershire sauce. Fold in the chopped egg and season the dressing with salt and pepper. Refrigerate until chilled.
Meanwhile, in a large saucepan, cover the bratwursts and onion with the beer and 3 cups of water and bring to a boil. Simmer over moderately low heat until an instant-read thermometer inserted in the center of each brat registers 165°, 8 to 10 minutes. Transfer the brats to a plate.
Heat a grill pan or large cast-iron skillet. Cook the bratwursts over moderately high heat until browned all over, about 5 minutes. Fill the rolls with the pastrami, cheese and brats. Top with some of the dressing and giardiniera and serve.
Lots of breweries in the US are making Belgian-style beers that go expertly with brats.
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