1) Make the polenta and set it aside to stay warm.
2) Heat the olive oil in a skillet over medium—high heat and add the country ham. Cook until the pieces start to curl up a little. Remove the ham from the pan and set it aside for later. Reserve the cooking fat in the skillet.
3) Season the shrimp with salt and pepper before adding them to the hot skillet. Cook the shrimp until they are bright pink, about 2 minutes. Set the shrimp aside with the ham.
4) Turn the heat down and add the mushrooms and 1 tablespoon of butter to the skillet. Cook until the mushrooms start to get tender—about 5 minutes—and then add garlic and cook them together until the garlic starts to brown. Add the chicken broth to the pan and make sure you scrape the bottom of skillet to get up all the good bits and flavors. Turn the heat up and cook until the broth reduces by about half.
5) When the broth is reduced, return the shrimp and ham to skillet. Add the remaining butter and cook (stirring often so that nothing sticks) until the sauce thickens, about 5 more minutes. Serve over the grits you cooked earlier and garnish with chopped green onions.
Directions for Polenta
Bring the milk, broth, water, salt, and pepper to a boil in a large, heavy saucepan over medium-high heat. Gradually whisk in the grits. Reduce the heat to low; simmer until the grits are smooth and thick, stirring constantly and tasting as you go to check the grits for softness. If they need more time, add water and keep stirring. It’s worth the effort.
When the grits are softened and ready, stir in the butter and cheese and serve