Make the potatoes: Preheat the oven to 350 degrees F. Toss the potatoes with 1 tablespoon olive oil and a generous pinch of salt. Arrange on a baking sheet; bake 35 to 40 minutes, or until tender when pierced with the tip of a knife. Remove and let cool.
Increase the oven temperature to 375 degrees F. In a cast-iron skillet, heat the remaining 3 tablespoons olive oil over medium heat until hot. On a flat surface, smash and break the potatoes with the back of a wooden spoon. Season with salt. Add to the skillet and cook until browned (do not stir them!), 4 to 5 minutes. Flip the potatoes and cook 3 to 5 additional minutes. Season with salt and sprinkle with the red wine vinegar.
Meanwhile, make the salmon: Line a rimmed baking sheet with foil; drizzle with 1 tablespoon olive oil. Arrange the salmon skin-side down on the baking sheet. Dot with the butter and drizzle with the remaining 1 tablespoon olive oil. Season thoroughly with salt. Use a fine-mesh strainer to sprinkle the cayenne over the fish. Bake, undisturbed, 6 to 8 minutes. (For well-done, bake an additional 5 minutes.) Remove from the oven.
Make the salad: Toss the parsley, tarragon, arugula and radishes. Serve the salmon and potatoes with the herb salad. Squeeze the lemon juice over the fish.