Brine the turkey: Combine the beer, mustard and 1 teaspoon each salt and pepper in a 1-gallon resealable plastic bag. Add the turkey, seal the bag and place in a large bowl; refrigerate 45 minutes.
Meanwhile, prepare the pesto: Roast the bell peppers on a grill, directly over the flame of a gas burner or under a preheated broiler, turning with tongs, until blackened. Seal the peppers in a plastic bag and let cool at least 10 minutes. Peel the peppers and discard the stems and seeds (do not rinse).
Pulse the peppers, basil, garlic, parmesan, pine nuts, red pepper flakes, anchovy paste, 1/2 teaspoon salt and 1 teaspoon pepper in a food processor until combined.
With the motor running, slowly add the olive oil through the feed tube. Adjust the seasoning if necessary. Refrigerate the pesto if not using immediately.
Preheat the oven to 375 degrees F. Remove the turkey from the brine (do not rinse) and place on a rack in a shallow roasting pan. Roast until a thermometer inserted into the thickest part of the breast registers 165 degrees F, about 1 hour. Transfer to a cutting board to cool.
Assemble the sandwich: Remove the meat from the bone and slice on a diagonal into 1/2-inch-thick pieces. Roughly chop the marinated vegetables and mix with 1/4 cup pesto in a bowl. Slice the bread in half horizontally to make a giant bun. Pull some of the bread out of the inside of the top half to hollow it out. Spread pesto on both halves. Layer the lettuce, provolone, turkey and vegetable-pesto mixture on the bottom half of the bread, then cover with the top half. Wrap tightly with plastic wrap to squeeze the sandwich together. Refrigerate 20 minutes, then slice into pieces.