Place the peeled potato chunks in a large pot and cover with cold salted water. Bring to a boil, then reduce to a simmer and cook for 15 to 20 minutes, until the potatoes are fork-tender.

While the potatoes cook, season the chicken on both sides with 1 teaspoon salt, 1/2 teaspoon black pepper, and the paprika. Lightly coat the chicken with the flour, shaking off any excess.

Heat the olive oil in a large skillet over medium-high heat. Add the chicken and cook for 5 to 7 minutes per side, until golden brown and cooked through. Transfer to a plate and set aside.

Reduce the heat to medium and add the butter to the same skillet. Add the minced garlic and cook for 30 seconds to 1 minute, stirring constantly, until fragrant.

Pour in the chicken broth and scrape up any browned bits from the bottom of the pan. Stir in the heavy cream, Dijon mustard, thyme, remaining salt, and remaining black pepper. Simmer for 3 to 5 minutes, until the gravy begins to thicken.

If using, stir in the Parmesan cheese and chopped parsley. Return the chicken to the skillet and spoon the gravy over the top. Simmer for 2 to 3 minutes to warm the chicken through and finish the sauce.

Drain the cooked potatoes well, then return them to the pot. Add the butter and warm milk or half-and-half. Mash until smooth and creamy, adding more milk if needed. Season with salt and black pepper to taste.

Serve the creamy garlic chicken over the mashed potatoes and spoon plenty of gravy over the top. Garnish with extra parsley if desired.