Heat 1 tablespoon of sesame oil in a large Dutch oven or soup pot over medium-high heat. Add the chicken pieces and cook until golden brown on all sides, about 5-6 minutes. The chicken doesn't need to be fully cooked through at this stage. Transfer to a plate and set aside.
Add the remaining tablespoon of sesame oil to the same pot. Add the white parts of the green onions, ginger, and garlic. Sauté for 2 minutes until fragrant, stirring constantly to prevent burning.
Pour in the chicken broth and water, scraping the bottom of the pot to release any browned bits. Add the rehydrated mushrooms, carrots, and bell pepper. Bring to a gentle boil, then reduce heat and simmer for 15 minutes.
While the soup simmers, whisk together the hoisin sauce, rice vinegar, soy sauce, brown sugar, and chili paste in a small bowl.
Return the chicken to the pot and simmer for another 5 minutes until fully cooked. Stir in the General Tso's sauce mixture and bring to a gentle boil.
Add the cornstarch slurry, stirring constantly until the soup slightly thickens, about 2 minutes.
Turn off the heat and stir in the green parts of the green onions. Taste and adjust seasonings as needed.
Serve over rice or rice noodles. Garnish with fresh cilantro, a sprinkle of sesame seeds, and a squeeze of lime.