Season both sides of the chicken breasts with half the salt, pepper, smoked paprika, and dried thyme. If breasts are uneven, gently pound to about 3/4 inch thickness for even cooking.
Heat a large 12-inch skillet over medium-high. Add 1 tablespoon olive oil. When shimmering, add chicken and sear without moving until well browned, about 4–5 minutes. Flip and brown the other side 3–4 minutes. Transfer chicken to a plate and tent loosely with foil.
Reduce heat to medium and add remaining 1 tablespoon olive oil and 2 tablespoons butter to the skillet. Add the halved potatoes cut-side down and sprinkle with the remaining salt. Cook undisturbed until the cut sides are nicely golden, about 8–10 minutes.
Stir the garlic and lemon zest into the potatoes and cook 30–45 seconds until fragrant. Scrape up any browned bits from the pan with a wooden spoon.
Pour in the chicken broth to deglaze, bring to a simmer, then return the chicken to the skillet, nestling it among the potatoes. Spoon a little liquid over each piece.
Reduce heat to low, cover the skillet, and simmer gently until the chicken reaches 165°F and potatoes are tender when pierced, about 8–10 minutes.
Uncover and add the remaining 2 tablespoons butter. Increase heat to medium-high for 1–2 minutes, spooning the melted butter and pan sauce over the chicken to glaze and thicken the sauce slightly.
Remove skillet from heat, sprinkle with chopped parsley, and let the dish rest in the pan for 5 minutes so juices redistribute and sauce settles.
Serve hot, spooning the garlic butter potatoes and pan sauce over each chicken serving.