Herbed mayo offers a cool counterpoint to the smoky rub patted onto the fillets.
Heat grill to medium-high; brush grates with vegetable oil. Rub both sides of salmon with olive oil. Stir together paprika, brown sugar, 1 teaspoon coarse salt, and 1 teaspoon pepper; rub over both sides of fish. Place fish, skin-side down, on grill and cook, flipping once, until just cooked through, 2 to 3 minutes a side. Transfer to a plate.
Stir together herbs and mayonnaise. Grill rolls, cut-sides down, until just toasted, about 1 minute. Spread cut sides with herbed mayonnaise. Divide salmon among rolls; top with lettuce, tomato, and cucumber. Season with flaky salt, pepper, and a squeeze of lemon; serve.