Put the evenly cut potatoes in the bottom of a clean 4- to 6-quart slow cooker. Keep the chicken refrigerated until you are ready to add it. Check your appliance instructions for its recommended fill level.

Stir together the broth, garlic, lemon zest, 1 tablespoon lemon juice, oregano, salt, and pepper. Pour it over the potatoes, then place the chicken thighs on top and distribute the butter over the chicken.

Cover and cook on LOW for about 4–5 hours or HIGH for 2½–3½ hours. Slow cookers vary, so begin checking near the shorter time. The potatoes should yield easily to a fork and the chicken must reach at least 165°F (74°C).

If the chicken is ready before the potatoes, transfer it to a covered dish and keep hot at 140°F (60°C) or above. Continue cooking the potatoes until tender. Do not leave cooked chicken sitting at room temperature while they finish.

Return any removed chicken to the cooker. Stir the remaining 2 tablespoons lemon juice into the broth and sprinkle with parsley. Taste before adding more salt; the broth and chicken juices can vary in saltiness.

Divide the chicken and potatoes among four plates and spoon the lemon-garlic broth over them. Serve promptly with a separate vegetable side if you like. This is a brothy meal rather than a thick gravy.