Scrub the baby potatoes and trim the green beans. Thinly slice the onion, mince the garlic, and cut the lemon in half; juice one half and thinly slice the other half for serving.

Add the potatoes, onion, garlic, chicken broth, olive oil, half the salt, half the pepper, and dried thyme to the slow cooker. Stir once, then cover and cook on HIGH for 2 to 3 hours for the listed cook time, or on LOW for 4 to 5 hours, until the potatoes are nearly tender.

Nestle the green beans on top, cover again, and cook 20 to 30 minutes more on HIGH or 30 to 45 minutes on LOW, until the vegetables are tender-crisp.

Pat the fish fillets dry and season with the remaining salt and pepper. Lay the fish in a single layer on top of the vegetables, spooning a little cooking liquid over the fish. Cover and cook on HIGH for 20 to 30 minutes or on LOW for 30 to 40 minutes, just until the fish is opaque, flakes easily, and reaches 145°F / 63°C. If using LOW, the total cook time will be about 5 to 6 1/2 hours.

Turn off the slow cooker. Add the butter and lemon juice, then gently spoon the sauce over the fish. Sprinkle with parsley, top with the reserved lemon slices, and serve right away with the potatoes and green beans. For food safety, keep the slow cooker covered as much as possible and refrigerate leftovers within 2 hours.