For grits: Bring 3 cups water to a simmer in a large saucepan. Gradually whisk in grits. Turn heat to low; gently simmer until grits begin to thicken. Continue cooking, stirring often and adding water by 1/4 cupfuls if too thick, until tender, about 1 hour. Stir in cheese, butter, and jalapeño, then cream. Season with salt and pepper. Keep warm.
For shrimp: Meanwhile, heat a large heavy skillet over medium heat. Add tasso; sauté until fat begins to render, about 5 minutes (if tasso is very lean, add 1 tablespoon oil to skillet). Add garlic and 1 tablespoon butter; stir until butter melts. Add shrimp. When garlic begins to brown, add beer and chicken stock. Simmer until shrimp is cooked through, about 2 minutes. Remove skillet from heat; set aside.
Heat a large nonstick skillet over medium heat. Add remaining 1 tablespoom butter to skillet; swirl to melt and cover bottom of pan. Crack eggs into pan and cook until whites are just set but yolks are still runny, about 3 minutes.
Divide grits among bowls, forming a well in center. Spoon shrimp mixture into center of grits. Top with egg. Sprinkle tarragon over.