To make the grits: Bring 4 cups water to a boil in a 4-quart heavy saucepan and stir in grits and salt. Return to a boil, stirring frequently.
Reduce heat and simmer grits, uncovered, stirring occasionally, until very thick and most of water is absorbed, about 10 minutes.
Reduce heat further and cook grits at a bare simmer until cooked through, about 1 hour 10 minutes, stirring frequently and adding remaining 1 1/2 cups water (1/4 cup at a time) whenever grits become very thick.
Add tasso, butter, and salt and pepper to taste, stirring until butter is melted.
Meanwhile, make the poached eggs: Combine the water, vinegar, and salt, to taste, in a large skillet and bring to a gentle simmer.
Crack the eggs and slide them into the skillet. Poach the eggs, turning occasionally with a spoon, until firm and slightly cooked through, about 4 minutes. Using a slotted spoon, remove the eggs and transfer to a kitchen towel, to drain.
Divide the grits among plates and top with the poached eggs. Serve immediately.