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Combine broth, water and milk in a medium pot; cover and bring to a light boil. Slowly stir in grits; cook, covered and stirring occasionally, until thickened and tender, 10 to 15 minutes. (Continue cooking for up to 30 minutes for thicker consistency, if desired.) Stir in cheese, salt and pepper.
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Meanwhile, using your hands, break mushrooms into large pieces. Heat oil in a large skillet over medium heat. Add the mushrooms and cook, undisturbed, until golden, 3 to 5 minutes. Stir in garlic; cook, stirring, for 2 to 3 minutes. Add soy sauce; cook, stirring, until the mushrooms are a deep golden color, about 1 minute.
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Divide the grits among 4 plates. Top each with some of the mushrooms and 1 egg. Sprinkle the eggs with pepper and scallions, if desired.