Pound the green chilies with salt until smooth.
Add galangal, lemongrass, lime zest, coriander roots, and white peppercorns; pound until well combined.
Add shallots, garlic, coriander seeds, cumin, and shrimp paste; pound everything into a uniform, smooth paste.
Store in an airtight container. It holds up in the refrigerator for up to one week.
Cut the winter melon into bite-sized pieces. Cut the chicken (breast or thighs) into bite-sized pieces.
Heat the coconut cream in a pot over medium heat with the curry paste until fragrant and the oil begins to separate.
Add the chicken and stir-fry until the surface turns opaque. Then add the winter melon and pour in the coconut milk.
Season with fish sauce and palm sugar.
Simmer until the chicken is tender and the melon is fully cooked. Taste and adjust seasoning as needed.
Turn off the heat. Garnish with sweet basil leaves and sliced red chili.
Serve hot with Thai rice noodles (Khanom Jeen) and fresh vegetables on the side.