In a medium saucepan, combine the water, milk, 1/2 teaspoon kosher salt, and a few grinds of black pepper. Bring the mixture to a boil. Gradually whisk in the grits, ensuring a smooth incorporation. Lower the heat to a gentle simmer and cook, stirring frequently, until the mixture thickens, approximately 5 to 7 minutes. Fold in the grated cheese until fully melted. Adjust seasoning with additional salt and pepper as needed, and keep the grits warm over low heat.

In a large nonstick skillet, melt 2 tablespoons of butter over medium heat. Add the thinly sliced carrots and cook for about 2 minutes until they begin to soften. Introduce the sugar snap peas and the scallion pieces, seasoning lightly with salt. Stir well and pour in 2 tablespoons of water. Cover the skillet and steam the vegetables until they reach a crisp-tender texture, about 4 to 5 minutes. Stir in the whole-grain mustard, mixing thoroughly. Transfer the vegetables to a bowl and cover to keep warm.

Wipe the skillet clean and return it to medium heat. Melt the remaining tablespoon of butter. Carefully crack the eggs into the skillet and season with salt and pepper. Cook until the whites begin to set, approximately 3 minutes. Cover the skillet and continue cooking for 1 to 2 more minutes, ensuring the whites are fully set while the yolks remain runny.

To assemble, spoon the warm grits into serving bowls. Top each portion with a fried egg, a generous helping of the sautéed vegetables, and a sprinkle of parsley, dill, and thinly sliced scallion. Serve immediately.