Heat olive oil in a large skillet over medium-high heat. Tilt the pan so the oil coats the entire surface.
Season chicken legs with salt and pepper to taste and sear until browned on both sides. This took about 4-5 minutes on our stovetop. Transfer to a plate and cover with foil.
Lower the heat to medium and add more olive oil. Add garlic and cook for 1-2 minutes until fragrant and golden.
In a measuring cup with a pour spout, whisk together broth, flour, one tablespoon of dill, lemon juice, lemon zest, and mustard.
Add the mixture to the pan and whisk briskly until thickened and bubbling. This should take about 5 minutes.
Place the chicken back in the skillet and reduce the heat to low. Simmer until the chicken is thoroughly cooked, about 8 minutes on our stovetop.
Serve with any sides you'd like, though vegetables and rice rounds out this dinner beautifully. Be sure to use the sauce as a kind of gravy!