Combine water and 1/2 tablespoon kosher salt in a medium, heavy saucepan; bring to a boil. Whisk in grits, and lower heat to a simmer. Cook 20 minutes or until almost tender, stirring occasionally. Remove from heat, and add cheese, butter, pepper to taste (about 1/8 teaspoon), and Tabasco to taste (about 1/8 teaspoon), whisking to combine thoroughly. Cover and keep warm.
Sauté mushrooms in hot oil in a sauté; pan or large skillet over medium-high heat about 3 minutes. Add shallot and country ham, and cook 30 seconds. Add port wine and chicken stock, and cook until mixture is reduced by three-fourths.
Spoon grits onto serving plates, and top with poached eggs. Spoon ham and mushroom sauce evenly over the eggs. Toss frisée with vinaigrette, and season with salt and pepper; scatter frisée mixture along the edges of the grits, and serve.