Pat the trout fillets dry. Season on both sides with the kosher salt and black pepper.
Place the flour in a shallow dish. Dredge each fillet lightly in flour, shaking off excess; set on a rack or plate.
Heat a 12-inch skillet over medium heat. Add the vegetable oil and 2 tablespoons unsalted butter; when the butter foams, add the trout fillets skin-side down.
Cook until the skin is crisp and the edges turn opaque, 3β4 minutes. Flip and cook until just opaque and flakes easily, 2β3 minutes more. Transfer fillets to warm plates. Wipe out the skillet.
Return the skillet to medium heat and add 6 tablespoons unsalted butter. Cook, swirling, until the butter turns deep golden brown and smells nutty, 3β5 minutes.
Stir in the sliced almonds and toast in the butter until lightly golden, 1β2 minutes. Remove from heat; stir in the lemon juice and Worcestershire sauce (it will sizzle), then the parsley.
Spoon the almond brown-butter sauce over the trout. Serve immediately with lemon wedges.