Season both sides of the fish with salt and pepper, then coat evenly in the flour.
Heat a sauté pan over medium-high heat. Gently place the fish in the pan, skin side down, and cook until golden brown. Carefully flip the fish to avoid breaking it, and continue cooking until the other side is golden brown and the internal temperature reaches 145°F. Transfer the fish to a plate.
In the same pan, add the remaining butter and swirl it over medium-high heat until it turns a very light brown.
Add the garlic and shallot to the pan, cooking until the butter develops a rich brown color. Stir in the chopped parsley at the last moment.
Squeeze fresh lemon juice over the fish and pour the hot butter sauce on top.
Toss the salad ingredients in a bowl, then serve alongside the fish.