Heat a gas grill over high for about 10 minutes.
Meanwhile, in a small bowl, combine ½ cup mayonnaise, ¼ cup minced cilantro, 2½ tablespoons mustard, and 4 teaspoons miso; mix until thoroughly combined. Set aside.
Form beef into four 5-inch-wide and ½-inch thick patties (6 ounces each). Place on a baking sheet and set aside.
Remove crown and skin from pineapple, then cut eight ⅛-inch thick slices of pineapple rounds. Using a small round cookie cutter, or carefully with a paring knife, cut out tough core from each slice. Arrange slices on a separate baking sheet and lightly coat with a splash of oil.
Over high heat, place pineapple slices on grill grate and grill until char marks form and pineapple is softened, 6–8 minute, flipping every 2 minutes. Transfer to baking sheet and set aside.
Season one side of burger patties with 1 teaspoon kosher salt (about ¼ teaspoon each); place patties, seasoned side down, on grate. Season tops of patties with another teaspoon of salt (¼ teaspoon per patty) and grill until charred and easily release from grate, about 4 minutes. Flip patties, place a slice of Pepper Jack on each patty and continue grilling until cheese is melted and burgers register 125°F (52°C) for medium-rare or 135°F (57°C) for medium, 1–2 minutes more. Transfer burgers to a large plate.
Toast buns, cut side down, on grill until golden brown, about 2 minutes.
To assemble burgers, spread about 1 tablespoon of sauce onto each cut side of both top and bottom buns. On bottom buns place a burger patty followed by 2 pineapple slices and 1 slice of iceberg lettuce. Serve immediately.