Bring water to a boil in a medium saucepan. Stir in oats and dash of salt. Reduce heat; simmer 5 minutes, stirring occasionally.
Heat olive oil in a skillet over medium heat; add mushrooms to pan. Sauté 4 minutes. Stir 1 1/2 tablespoons crumbled goat cheese, half-and-half, 1/8 teaspoon salt, and thyme into warm oatmeal. Top with mushrooms and 1 1/2 teaspoons crumbled goat cheese.