Heat 1 teaspoon sesame oil in a large soup pot over medium heat; cook and stir shrimp until bright pink, about 5 minutes. Transfer shrimp to a plate.
Heat remaining 1 teaspoon sesame oil; cook and stir carrots and sliced mushrooms until they begin to soften, about 5 minutes. Pour in broth, hoisin, soy sauce, and chile-garlic sauce. Bring to a boil; boil 3 minutes.
Add asparagus and ramen noodles; boil 3 minutes. Turn off heat. Stir shrimp back into the pot; let stand 4 minutes.
Divide mixture between two bowls. Garnish with chopped green onions.