In a small bowl, toss the carrot and vinegar. In a saucepan, boil the stock and ginger until reduced to 1 1/4 cups, 15 minutes. Remove from the heat; whisk in the sesame oil and hoisin.
Meanwhile, in a large skillet, heat the vegetable oil over medium-high. Cook the mushrooms, stirring occasionally, until golden brown, 8 to 10 minutes; season with salt and pepper.
Cook the ramen according to package directions. Drain; divide among 4 bowls. Top with the ginger sauce, carrot, mushrooms, bean sprouts and chili-garlic sauce; garnish with the sesame seeds and sliced scallions.