Finely chop 1 small shallot (about 2 tablespoons) and mince 2 garlic cloves. Finely grate the zest of 1 large lemon (about 1 tablespoon). Halve the lemon and cut 4 thin slices from one half; set the slices aside for garnish. Juice the remaining lemon half (about 3 tablespoons). Pick the fronds from ¼ small bunch fresh dill and finely chop until you have 2 tablespoons.

Pat 4 (6-ounce) salmon fillets dry with paper towels. Season all over with 1 teaspoon of the kosher salt and ¼ teaspoon of the black pepper.

Heat 1 tablespoon olive oil in a large nonstick frying pan over medium-high heat until shimmering. Add the salmon skin-side down and sear undisturbed until golden-brown on the bottom, about 4 minutes. Flip and continue to cook until an instant-read thermometer inserted into the middle of the thickest fillet registers 120°F to 130°F for medium-rare, or 135°F to 145°F if you prefer it more well-done, 2 to 4 minutes more. Transfer skin-side up to a plate.

Reduce the heat to low and add 2 tablespoons unsalted butter. When melted, add the shallot, garlic, remaining ½ teaspoon kosher salt, and remaining ¼ teaspoon black pepper. Cook, stirring occasionally, until just starting to soften, about 30 seconds. Add 2 tablespoons all-purpose flour, stir to evenly coat the shallots and garlic, and cook for 1 minute more.

Add 1 cup half-and-half and ½ cup low-sodium chicken or vegetable broth, and whisk until smooth. Bring to a simmer over medium-low heat. Simmer until the sauce is thickened enough that it coats the back of a spoon, about 1 minute. Add the lemon zest, lemon juice, and dill, and stir to combine. Return the salmon to the pan skin-side up, nestling the pieces in the sauce. Simmer for 1 minute to meld the flavors. Garnish each piece with a reserved lemon slice and serve immediately.