Make the spice rub: In a small bowl combine paprika (i2), smoked paprika (i3), garlic powder (i4), onion powder (i5), dried thyme (i6), dried oregano (i7), cayenne pepper (i8), 1 teaspoon kosher salt (i9), and 1/2 teaspoon black pepper (i10). Stir until evenly mixed.

Prepare the potatoes: Place peeled, cut potatoes (i15) in a large pot and cover with cold water by about 1 inch. Add 1 teaspoon kosher salt (i9). Bring to a boil over high heat, then reduce to a simmer and cook until potatoes are tender when pierced with a fork, about 12–15 minutes depending on size.

Warm the dairy for the mash: While potatoes cook, warm the milk (i16) and heavy cream (i17) together in a small saucepan or microwave until hot but not boiling. Keep warm.

Season the salmon: Pat salmon fillets (i1) dry with paper towels. Evenly rub the spice mixture over the flesh side of each fillet, pressing gently to adhere. Season the skin side (or opposite side if skinless) with the remaining 1/2 teaspoon black pepper (i10) and a light pinch of salt (i9).

Blanch the green beans: Bring a medium pot of salted water to a rolling boil. Add green beans (i20) and cook for 2–3 minutes until bright green and just tender. Immediately transfer beans to an ice-water bath to stop cooking and preserve color. Drain and set aside.

Finish the mashed potatoes: Drain potatoes (i15) thoroughly and return to the hot pot off the heat for a minute to allow excess moisture to evaporate. Add 4 tablespoons butter (i18), then mash with a potato masher or ricer. Gradually stir in warmed milk (i16) and cream (i17) until the mash reaches your preferred consistency. Season with salt and pepper to taste (i19) and keep warm while you cook the salmon.

Sear the salmon: Heat a large heavy skillet (equipment) over medium-high heat until very hot. Add neutral oil (i11) and swirl to coat. Place salmon fillets (i1) in the pan, flesh-side down if skinless or skin-side down if skin-on depending on preference; press gently so the fillets make even contact with the skillet. Cook undisturbed until a dark crust forms, about 3–4 minutes. Flip carefully and cook the second side for 2–3 minutes for medium doneness. Remove salmon to a warm plate and tent loosely with foil.

Make the brown-butter-lemon sauce: In the same skillet over medium heat, reduce heat to medium-low and add 4 tablespoons butter (i12). Swirl the pan as the butter foams and then turns light brown and nutty, about 2–3 minutes. Remove from heat, stir in lemon juice (i13) and 1 tablespoon chopped parsley (i14) if using, and season with a pinch of salt (i9) and a grind of black pepper (i10). Spoon sauce over the salmon fillets.

Sauté the green beans: In a separate large skillet over medium heat, add 1 tablespoon olive oil (i21). Add sliced or minced garlic (i22) and red pepper flakes (i23) and cook for 30–45 seconds until fragrant. Add the blanched, drained green beans (i20), sprinkle with 1/4 teaspoon kosher salt (i24) and cracked black pepper (i25), and toss to coat. Cook, stirring occasionally, for 2–3 minutes until beans are heated through and glossy. Add 2 tablespoons butter (i26) at the end if you want extra richness and toss until melted.

Plate and garnish: Spoon a generous mound of mashed potatoes on each plate, arrange a salmon fillet on top or just beside the potatoes, and add a portion of green beans. Drizzle brown-butter-lemon sauce around the salmon and potatoes, sprinkle with additional chopped parsley or chives (i28), and serve with lemon wedges (i27).

Check doneness and serve immediately: Salmon is best served right away while the crust is crisp and the center tender. If you prefer a firmer texture, cook a minute or two longer on the second side and recheck the internal temperature.

Make-ahead notes: Mashed potatoes can be made up to one day ahead and gently reheated with a splash of milk and a pat of butter. Green beans can be blanched and chilled ahead of time, then quickly finished in the skillet just before serving.