Bring water, broth, 2 tablespoons butter, and salt to a boil in a heavy saucepan. Stir in grits. Reduce heat to low. Cover and simmer, stirring occasionally, until water is absorbed and grits are a thick consistency, about 15 minutes. Remove from heat.
Preheat oven to 325 degrees F (165 degrees C). Lightly oil a 8x10-inch baking dish.
Heat a skillet over medium-high heat. Add chorizo; cook and stir, breaking into small pieces, until crisp, 8 to 10 minutes. Remove from pan with a slotted spoon and drain on paper towels.
Heat 1/4 cup butter in a pot. Add shallot; cook and stir until shallot softens and butter is bubbly, 2 to 3 minutes. Add lobster and garlic. Cook just until lobster turns opaque, 2 to 3 minutes. Remove from heat.
Whisk shredded Gouda cheese and Cheddar cheese into the grits. Add chorizo, salt, and pepper. Pour into the prepared baking dish. Top with lobster, shallots, and pan drippings.
Bake in the preheated oven until heated through and bubbly, 15 to 20 minutes.