Preheat oven to 425°F. Rub 2 teaspoons of the oil all over pork. Combine thyme, sage, rosemary, 1/2 teaspoon of the salt, and 1/4 teaspoon of the pepper in a small bowl; rub herb mixture into pork to coat. Heat 1 teaspoon of the oil in a large skillet over high. Sear pork until browned on all sides, about 2 minutes per side. Transfer to a baking sheet; roast until a meat thermometer inserted into thickest part registers 140°F, 10 to 12 minutes. Let rest on a cutting board 5 minutes before slicing.
Meanwhile, add mushrooms and 1 tablespoon of the butter to skillet over medium-high heat. Cook, undisturbed, until browned on 1 side, 4 to 5 minutes. Stir, and cook, stirring often, until browned on all sides, 2 to 3 minutes. Add garlic and remaining 1 tablespoon butter, 1/2 teaspoon of the salt, and 1/4 teaspoon of the pepper; sauté until fragrant, about 1 minute. Stir in wine; cook until liquid evaporates. Whisk together broth and flour in a small bowl; stir into mushrooms, and cook, stirring occasionally, until sauce has thickened, about 5 minutes. Transfer to a bowl.
Heat remaining 3 teaspoons oil in a large skillet over medium-high. Add cabbage; cook, stirring often, until slightly wilted, 3 to 4 minutes. Whisk together vinegar, honey, and mustard in a small bowl; stir into cabbage. Top pork with mushroom sauce, and serve with cabbage.