Heat the 1 tablespoon unsalted butter and 1 tablespoon olive oil in a large pot over medium-high heat.
Add the ½ pound boneless, skinless chicken breast in a single layer without overcrowding (Note 8). Cook for about 2–3 minutes per side until lightly golden. The chicken does not need to be fully cooked at this stage. Remove to a plate and set aside.
Reduce heat to medium. Add the 1 small diced yellow onion and cook for 2–3 minutes until softened. Stir in the 4 cloves minced garlic and 2 tablespoons freshly grated ginger and cook for 30–60 seconds, just until fragrant. Do not allow the garlic to brown.
Stir in the 1 teaspoon lemon zest and 1½-2 tablespoons red curry paste. Cook for 1–2 minutes, stirring constantly, until the paste darkens slightly and becomes very fragrant. This step deepens the flavor of the soup.
Pour in the 4 cups low-sodium chicken broth, shaken 14 oz full-fat coconut milk, and 1 tablespoon fish sauce. Stir well, scraping up any browned bits from the bottom of the pot. Bring to a gentle simmer. Avoid a rolling boil (Note 9).
Return the chicken to the pot along with the sliced 8 ounces fresh shiitake mushrooms. Simmer gently for 6–8 minutes, or until the chicken reaches 165°F and is just cooked through (Note 10). Do not overcook. The chicken will continue cooking slightly from residual heat.
Stir in the sliced scallion. Taste and adjust with salt and pepper as needed.If you prefer a thinner broth, add ½–1 cup additional chicken broth. For a slightly richer texture, simmer uncovered for a few extra minutes (Note 11). Ladle into bowls and top with fresh cilantro, additional scallions, and a generous squeeze of lime. Serve with extra lime wedges on the side.