Pat the chicken pieces dry and season lightly with salt and pepper. Heat the oil in a large Dutch oven or deep skillet over medium-high heat. Add the chicken in a single layer and sear until just golden on the outside, about 2 minutes per side; you don't need to cook through. Transfer the chicken to a plate and set aside.
Reduce heat to medium. Add the chopped onion to the same pot and cook, stirring occasionally, until softened, about 4 minutes. Stir in the ginger, garlic, and minced lemongrass and cook another 45–60 seconds until fragrant.
Add the red curry paste to the aromatics and cook, stirring, for 1 minute to bloom the spices. Pour in about 1/2 cup of the chicken broth and scrape any brown bits from the bottom of the pot to deglaze.
Stir in the coconut milk and remaining chicken broth, then return the seared chicken to the pot. Bring to a gentle simmer and cook uncovered for 7–8 minutes so the chicken finishes cooking and flavors meld.
Add the mushrooms and sliced red pepper, simmer another 3–4 minutes until vegetables are tender-crisp. Taste and season with fish sauce, brown sugar, and lime juice — adjust to balance savory, sweet, and tangy.
Turn off the heat and stir in most of the cilantro and basil, reserving a little for garnish. Let the soup sit 2 minutes to allow the herbs to wilt slightly and flavors to settle.
Ladle the soup into bowls and top with green onions, remaining herbs, a squeeze of lime, and chopped peanuts if using. Serve immediately with steamed rice or rice noodles if desired.