Pat the chicken thighs completely dry with paper towels, then season both sides with salt and pepper. Heat olive oil in your skillet over medium-high heat until it shimmers before adding the chicken. Sear for 6 to 8 minutes per side until deeply golden, then transfer to a plate that won't be fully cooked yet, and that's exactly what you want at this stage.
Lower the heat to medium and melt butter in the same skillet you used for the chicken. Toss in the minced garlic and let it cook for about 15 seconds until fragrant but not browned. Sprinkle the flour over the garlic and stir constantly for around 30 seconds, letting it turn a light golden color to cook out the raw flour taste.
Pour in the chicken broth, lemon juice, half the chopped dill, and the remaining salt. Use your wooden spoon to scrape up all those browned bits stuck to the pan that's flavor you don't want to waste. Bring everything to a lively simmer, stirring often so the sauce thickens evenly without any lumps forming.
Slide the seared chicken back into the skillet in a single layer, along with any juices that collected on the plate. Let it cook for 4 to 6 minutes, until the sauce clings nicely to the chicken and the thickest piece reaches a safe internal temperature. This final simmer is where the chicken really soaks up all that lemony flavor.
Take the skillet off the heat and stir in the remaining fresh dill for one last burst of freshness. Give the sauce a taste and adjust the salt if needed before serving. Let everything sit for a couple of minutes so the sauce settles and thickens slightly before you plate it up.