Step 1

Mix mayonnaise, lemon zest and juice, relish, capers, mustard, and garlic in a small bowl to combine; season with salt.

Step 2

Prepare a grill for medium-high heat. Whisk Old Bay seasoning, baking powder, cayenne, ½ cup flour, and ½ cup cornstarch in a medium bowl. Whisk remaining ¼ cup flour and ¼ cup cornstarch in another medium bowl.

Step 3

Set a deep cast-iron skillet or large Dutch oven fitted with thermometer on grill; pour in oil to come 1" up sides. Heat until thermometer registers 375°.

Step 4

Meanwhile, thinly slice cucumbers lengthwise. Toss in a third medium bowl with vinegar and gently massage with your hands to soften them slightly. Let sit, tossing occasionally, at room temperature at least 10 minutes.

Step 5

Lightly brush insides of rolls with butter and grill just until toasted, about 30 seconds. Transfer to a plate.

Step 6

Gently whisk egg and beer in a small bowl to combine, then gently whisk into Old Bay mixture in 3 additions. Working in 2 batches, season fish lightly with salt. Dredge in flour mixture, knocking off any excess, then dip into batter and lift up with a fork, letting excess drain off and wiping against side of bowl to eliminate any thick drips. (Batter should completely coat the fish without looking clumpy.)Carefully lower fish into oil and fry, turning halfway through, until deeply browned and an instant-read thermometer inserted into the thickest part of each piece registers 125°–130°, 7–9 minutes. Using a slotted spoon, transfer to a wire rack set inside a baking sheet.

Step 7

Drain most of vinegar from cucumbers and toss in dill. Build sandwiches with fried fish, cucumbers, lettuce, and tartar sauce.

Step 8

Do Ahead: Cucumbers (without dill) can be pickled 2 days ahead. Cover and chill.