Pulse ginger, garlic, scallions, thyme, paprika, allspice, cinnamon, cloves, cayenne, salt, pepper, and oil in the bowl of a food processor until finely chopped. Transfer to a large bowl; add shrimp and toss to coat. Cover and chill at least 8 hours and up to 1 day.
Bring milk, 3 cups water, 1 teaspoon salt, and a pinch of pepper to a boil in a medium saucepan. Slowly add grits, stirring constantly; reduce heat to low. Simmer, stirring often, until thickened and cooked through, 10 to 15 minutes. Remove from heat and stir in cheese and butter; adjust seasoning. Cover to keep warm, adding more water to loosen as needed.
Heat a grill over medium-high; brush grill grates lightly with oil. Add shrimp and cook, flipping once, until charred and just cooked through, 2 to 3 minutes total. Serve over grits with sliced scallions.