A warm brown-butter vinaigrette ties all the flavors in this hearty salad together.
Combine potatoes and 1 tablespoon kosher salt in a straight-sided skillet; add enough water to cover. Boil until potatoes are easily pierced with the tip of a knife, 10 to 12 minutes. With a slotted spoon, transfer to a bowl; cover to keep warm. Generously season fish with kosher salt and pepper. Gently lower into skillet, skin-side down, adding more water if fish is not fully submerged. Return to a boil; then remove from heat, cover, and let stand until fish is just cooked through, 5 to 7 minutes (depending on thickness of fillet). Transfer to a plate, skin-side up; cover to keep warm.
Melt butter in a medium saucepan over medium-high. When it boils, reduce heat to medium; simmer until foamy. Continue cooking, stirring occasionally and scraping bottom of pan, until foam subsides, butter turns golden brown with a nutty aroma, and milk solids separate into brown specks that sink to bottom, 2 to 7 minutes. Remove from heat and immediately whisk in shallot, Dijon, and vinegar (mixture will foam and splatter). Season generously with kosher salt and pepper; cover to keep warm.
Halve potatoes and toss with spinach, mint, and two-thirds of vinaigrette; divide evenly among 4 plates. Remove skin from fish; drizzle with remaining vinaigrette. Flake into several pieces, divide among salads, and serve, sprinkled with flaky salt.