If brining salmon, in a large bowl or container, combine 8 cups cold water (or enough to cover salmon) and 4 Tbsp. salt and stir to dissolve. Add salmon and let brine for 10 minutes. (If skipping brining step, lightly season both sides of salmon with salt.)

In a cold, large, nonstick skillet, arrange salmon fillets in a single layer flesh side down. Cook over high heat until you hear sizzling, about 1 minute, then reduce heat to medium and cover pan. Cook 5 minutes, then uncover pan.

Continue to cook, undisturbed, until browned and crispy, 3 to 4 minutes more. Flip salmon and cook until other side is browned and crispy, 3 to 5 minutes more. Transfer salmon to a large platter or baking sheet. Drain and discard fat; wipe out skillet.

In same skillet over medium heat, cook garlic, capers, and 1 Tbsp. butter, stirring occasionally, until butter melts, about 1 minute. Add broth, wine, lemon zest, lemon juice, and pepper. Bring to a simmer and cook, stirring occasionally, until sauce slightly thickens, 3 to 5 minutes.

Reduce heat to low. Whisk in remaining 4 Tbsp. butter, 1 Tbsp. at a time, vigorously whisking after each addition; don’t stop whisking until all butter is incorporated and sauce is glossy. Stir in dill.

Spoon sauce over salmon or return salmon to skillet to warm through.