Preheat oven to 450°. Place 1½ lb. small potatoes, halved, quartered if larger, on a rimmed baking sheet, drizzle with extra-virgin olive oil, and season with kosher salt. Toss to coat and arrange in a single layer. Roast until golden brown underneath, about 30 minutes.

Meanwhile, place ⅓ cup all-purpose flour on a plate. Season four 4–6-oz. cod fillets, patted dry with paper towels, all over with salt and freshly ground pepper. Lay one side of each fillet in flour to coat; shake off any excess and place, flour side up, on another plate. Heat remaining 2 Tbsp. extra-virgin olive oil in a large nonstick skillet over medium-high. Cook fish, flour side down, undisturbed, until golden brown and crisp underneath, 5–7 minutes. Turn; cook until just cooked through, about 1 minute. Transfer, flour side up, to a clean plate.

Add 1 shallot, finely chopped, and 3 Tbsp. unsalted butter to same pan. Cook, stirring occasionally, until shallot is slightly softened, 1–2 minutes. Add ¼ cup fresh lemon juice, 1 Tbsp. Dijon mustard, 1 tsp. honey, ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt, and 1 cup water. Cook, stirring often, until thickened, 6–8 minutes. Remove from heat; add ¼ cup tarragon leaves and remaining 2 Tbsp. unsalted butter and stir until sauce is emulsified. Season with salt.

Arrange fish, flour side up, in pan and spoon some sauce over. Season with pepper and scatter more tarragon leaves on top. Serve with potatoes on the side.