Combine 3 1/2 cups water, the milk, 1/2 teaspoon salt and a few grinds of pepper in a saucepan and bring to a boil. Slowly add the grits, whisking constantly. Reduce the heat to low; cook, stirring often, until thickened, 5 to 7 minutes. Stir in the cheese until melted; season with salt and pepper. Keep warm over very low heat.
Meanwhile, melt 2 tablespoons butter in a large nonstick skillet over medium heat. Add the carrots and cook until slightly softened, 2 minutes. Add the sugar snap peas and large scallion pieces; season with salt and toss. Add 2 tablespoons water, cover and simmer until the vegetables are crisp-tender, 4 to 5 minutes. Stir in the mustard. Transfer to a bowl and cover to keep warm.
Wipe out the skillet and melt the remaining 1 tablespoon butter over medium heat. Crack in the eggs; season with salt and pepper. Cook until the whites start setting, about 3 minutes, then cover and cook until the whites are set but the yolks are still runny, 1 to 2 more minutes. Top each serving of grits with a fried egg, the vegetables, parsley, dill and sliced scallion.