In a large saucepan, bring the broth, milk, butter to a boil. Stir in the grits, cooking according to the instructions on the packaging. Once cooked remove from the heat and stir in the parmesan cheese, season with salt and pepper to taste. Set aside with a lid to keep warm.
In a bowl, mix the shrimp, oil, old bay, scallions, garlic, and lemon juice. Season with salt and pepper and set aside for 15 minutes.
Heat a large non-stick pan over medium heat. Add the contents of the bowl of shrimp into the pan and sauté until the shrimps turn firm, pink and no longer translucent (internal temperature should be 125°F using a thermometer).
Serve the shrimps with the grits. Garnish with thyme sprigs.