Position racks in the upper and lower thirds of the oven; preheat to 425 degrees F. Toss the fries with 1 tablespoon olive oil on a baking sheet and sprinkle with the spiced salt. Bake on the lower oven rack, tossing halfway through, until golden, about 25 minutes.
Meanwhile, make the tartar sauce: Mix the mayonnaise, pickle and brine, scallions, parsley and a pinch of kosher salt in a small bowl.
Season the salmon with kosher salt and pepper. Heat the remaining 1 tablespoon olive oil in a large ovenproof nonstick skillet over high heat. Reduce the heat to medium; add the salmon and cook until golden brown on the bottom, 5 minutes. Transfer the skillet to the upper oven rack; bake until the salmon is just opaque, about 3 minutes.
Spread some tartar sauce on the rolls. Sandwich with the salmon (cut into pieces, if needed), lettuce and tomatoes. Serve with the fries and more tartar sauce for dipping.