Start the potatoes: Put the potatoes in a large pot, cover with cold salted water, and bring to a boil. Simmer for 15 to 20 minutes, until tender when a knife slides through a chunk without resistance. Drain well, then leave them in the hot pot for a minute so the steam escapes.
Season and sear the chicken: Pat the chicken dry, then season both sides with salt, pepper, and paprika. Heat the olive oil in a large pan over medium-high heat. Lay the chicken in the pan without crowding it and cook for 5 to 6 minutes on the first side, then flip and cook for another 5 minutes, until the juices run clear and the thickest part is cooked through. Move the chicken to a warm plate to rest while you make the sauce.
Make the garlic butter sauce: Turn the heat to medium and melt the butter in the same pan, scraping up the tasty brown bits from the bottom. Add the garlic and thyme and cook for about a minute, stirring, until it smells sweet and fragrant. Do not let it brown dark, or it will taste bitter. Spoon a little of the sauce over the resting chicken.
Mash the potatoes: Mash the drained potatoes with the butter first, then add the warm milk a little at a time, until smooth and soft but not runny. Season with salt and pepper. Warm milk matters here: cold milk cools the mash and can make it gluey.
Serve: Spoon the mash onto warm plates, slice the chicken or place it beside the mash, and pour the garlic butter over the top. Scatter over the parsley and serve with lemon wedges for anyone who wants them.