To make the broth, heat the oil in a large saucepan over medium heat. Add the garlic and onions and cook, stirring, until softened, about 5 minutes.
Stir in the lemongrass, soup base, and ginger and cook for 2-3 minutes.
Add the water, lime leaves, and chile and bring to a simmer. Cover and simmer for 1 hour. Add the button mushrooms for the last 10 minutes of cooking time. Remove the lime leaves. Adjust the seasoning.
Bring a large saucepan of salted water to a simmer. Add the chicken breasts and poach until cooked through, 8-10 minutes. Transfer to a plate, cool slightly, then shred with two forks or your hands.
To assemble the soup, divide the shredded chicken between bowls. Ladle in the broth with mushrooms and garnish with lime wedges, cilantro, cherry tomatoes and chiles.