1. Prepare the Polenta: 

In a saucepan, melt 1 stick of butter with a sprig of rosemary over medium heat until the butter foams and turns golden brown. Pour into a bowl, discard the rosemary, and set aside. 

Bring 4 cups of water to a boil in the same pan. Add 1 teaspoon salt, then gradually whisk in 1 cup polenta. Reduce heat to low and cook, stirring often, until creamy (20-25 minutes). Add a splash of water or milk if it thickens too much. 

Stir in the reserved brown butter and ½ cup grated Parmesan. Adjust seasoning and serve. 

2. Cook the Lamb Loin Chops: 

While the polenta is cooking, heat 2 tablespoons of olive oil in a large skillet over medium-high heat. Season the American lamb chops generously with salt, pepper, minced garlic, and rosemary. Once the oil is hot, add the lamb chops to the pan. Cook for about 3-4 minutes per side for medium-rare (or longer if you prefer them more done). Once cooked, transfer the lamb chops to a plate and cover loosely with foil to keep warm. 

3. Make the Red Wine Sauce: 

In the same skillet, reduce the heat to medium and add the shallots. Cook for 2-3 minutes until softened. Add the red wine, scraping up any brown bits from the bottom of the pan with a wooden spoon. Let the wine reduce by half, about 5 minutes. Add the broth and thyme, and simmer for another 5 minutes. Remove the thyme sprig and stir in the butter. Season with salt and pepper. Let the sauce thicken for a few more minutes. 

4. Serve: Spoon the creamy polenta onto plates. Place two lamb chops on top of the polenta and drizzle with the red wine sauce. Garnish with extra rosemary or fresh herbs if desired.