Pat trout fillets dry with paper towels. Sprinkle all over with 1 teaspoon salt and pepper. In a shallow dish such as a pie plate, spread cornstarch in an even layer. Working with 1 fillet at a time, dredge both sides in the cornstarch, pressing gently to adhere. Shake off excess cornstarch and transfer to a large plate.

In a 12-inch nonstick or cast iron skillet, heat 1 tablespoon butter over medium heat until just melted and beginning to bubble. Carefully place 2 fillets, skin side down, in skillet and use a thin spatula to gently press flat into the skillet; cook until lightly browned and crispy on the bottom, 2 to 4 minutes. Using 2 thin spatulas, gently flip fillets flesh side down and cook until flesh is opaque and bottom begins to turn golden, about 1 minute.

Transfer cooked fillets to a large serving platter, flesh side up. Repeat cooking with 1 more tablespoon butter and remaining 2 trout fillets.

Add remaining 6 tablespoons butter to the now-empty skillet. Cook, stirring frequently, until butter is light golden brown with a nutty, toasty aroma, 1 to 2 minutes. Stir in almonds and cook, swirling and stirring often, until almonds are deep golden brown and milk solids in butter have turned a hazelnut color, about 2 minutes.

Immediately remove skillet from heat and quickly but carefully swirl and stir in lemon zest, juice, parsley, and remaining ½ teaspoon salt, scraping up any browned bits from bottom of pan. Pour sauce over trout fillets. Serve immediately.