Put the onion in a small bowl, cover it with very hot water, and let it sit for 15 minutes. Roughly chop the garlic, sprinkle it with a generous pinch of salt, and mash it into a paste with the side of a chef’s knife. In a medium bowl, whisk the mashed garlic, 1 Tbs. oil, 2 Tbs. buttermilk, the lemon juice, dill, 1/4 tsp. salt, and a few grinds of pepper. Toss the cucumber in the bowl with the dressing. Drain the onion, toss it with the cucumber salad, and let sit to allow the flavors to meld.

Put the flour in a shallow bowl and, in another shallow bowl, mix the remaining 3/4 cup buttermilk with 1 tsp. salt. Season the chicken with 3/4 tsp. salt and 1/4 tsp. pepper. Dip the chicken in the buttermilk and then dredge it in the flour. (You can let the chicken sit in the flour while the oil heats; gently shake off excess flour before cooking.)

Choose a skillet (preferably cast iron) that’s large enough to fit the chicken. Pour in oil to a depth of 1/4 inch (about 1 cup for a 10-inch skillet or 1-1/4 cups for an 11-inch skillet). Heat over medium-high heat. When the oil is shimmering and the chicken sizzles briskly when a corner is dipped in the oil, cook the chicken until golden brown on both sides, 2 to 3 minutes per side. Transfer the chicken to paper towels and pat lightly to absorb excess oil. Sprinkle the chicken with a pinch of salt and serve it with the cucumber salad.