Heat 2 teaspoon vegetable oil in a large skillet over medium-low heat. Cook and stir onions, carrots, mushrooms, garlic, salt, and black pepper in the hot oil until coated, about 1 minute. Cover and cook until vegetables are soft, about 30 minutes. Uncover and increase heat to medium; cook and stir until vegetables are brown and caramelized, 7 to 10 more minutes.
Pour white wine into vegetables and simmer until wine is reduced by 2/3, 2 to 3 minutes. Stir in sun-dried tomatoes, marjoram, thyme, and water. Bring to a simmer, cover, and cook until vegetables are very soft, about 30 minutes. Strain liquid into a large measuring cup, using a spoon to press as much liquid as possible into the cup.
Heat 1 tablespoon vegetable oil in a saucepan over medium heat; stir in flour and cook, stirring constantly, until flour mixture is golden brown, about 5 minutes. Whisk in reserved vegetable liquid and simmer until thickened, 3 to 5 minutes.