Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add potatoes and cook, turning once, until brown and crispy, 5 to 6 minutes per side. Transfer to a plate and season with 1/4 teaspoon salt; cover with foil to keep warm.
Combine the remaining 1 tablespoon oil, 1/4 teaspoon salt, shallot and vinegar in a small saucepan. Bring to a boil over medium heat. Remove from the heat and whisk in buttermilk. Place salmon in a medium bowl and toss with the warm dressing. Divide arugula among 4 plates and top with the potatoes and salmon.