Preheat oil to 350˚F.
For lemon-relish spread: Place all ingredients into a mixing bowl and whisk together. Set aside until ready to use.
For quick pickled watermelon radish: Place vinegar, sugar and salt into a mixing bowl and stir together until sugar and salt dissolves. Add watermelon radish and stir together until completely submerged. Cover with plastic wrap and set aside for 30 minutes.
For beer battered shrimp: Line a large baking sheet with paper towels and top with a single layer of shrimp. Place a sheet of paper towel over the shrimp and gently press to absorb as much liquid as possible. Season the shrimp with salt and pepper.
Whisk together everything except for the beer. When ready to fry, stir in the beer until just incorporated and no dry spots remain.
Dip each shrimp into the cornstarch and shake off any excess. Then dip each into the beer batter, shaking off any excess (you want to make sure to have a very thin layer of batter fully encasing each dipped ingredient).
Carefully drop each battered shrimp into the oil, in batches (make sure not to over-crowd the pot/fryer which will make the oil temp drop, causing the food to become soggy and not crisp) and fry for 4 to 5 minutes or until golden brown. Drain onto paper towels for 30 seconds before transferring to a baking sheet, lined with a cooling rack. Season with salt and pepper. Repeat until all of the shrimp have been fried.
To assemble: Slather spread into each of the toasted and buttered brioche rolls and stuff with shrimp, watermelon radish pickles, and top with sliced chives. Serve with lemon wedges.