Preheat oven to 425°F. Pat trimmed chicken thighs dry with a paper towel; sprinkle both sides evenly with ¼ teaspoon each salt and pepper. Place the chicken, skin-side down, in a large cast-iron skillet. Cook over medium heat, undisturbed, until the skin is deeply golden, about 15 minutes. Flip and cook until golden on the other side, about 2 minutes. Transfer, skin-side up, to a plate; reserve the drippings in the pan.

Meanwhile, zest 1 lemon to yield ½ teaspoon zest; set aside. Slice the lemon into ¼-inch slices. Return the skillet with the drippings to medium-high heat. Add the lemon slices; cook, undisturbed, until they start to release juices and are browned on the bottom side, 1 to 2 minutes. Transfer the lemon slices to the plate with the chicken.

Add broccoli florets and the remaining ¼ teaspoon each salt and pepper to the pan; cook, stirring constantly, until bright green, about 1 minute. Remove from heat. Nestle the browned chicken thighs, skin-side up, into the pan with the broccoli, reserving lemon slices on the plate.

Roast until the broccoli is tender and slightly charred and an instant-read thermometer inserted into the thickest part of chicken registers 170°F, about 10 minutes. Transfer the broccoli and chicken to a serving platter; tent with foil to keep warm.

Return the skillet to medium-high heat. Add ⅔ cup chicken broth and 2 teaspoons mustard, whisking until combined; add the reserved lemon slices. Bring to a boil. Reduce heat to maintain a lively simmer; whisk in 1 tablespoon parsley, 1 teaspoon thyme and the reserved lemon zest. Remove from heat and whisk in butter pieces, 1 piece at a time, until fully incorporated. Drizzle the sauce over the chicken and broccoli. Garnish with additional thyme, if desired.