Discard 1 or 2 outer layers from lemongrass and trim root ends. Thinly slice enough of lower 6 inches of stalks to yield 1/2 cup, then finely grind in a blender with garlic, chile, turmeric, and 4 tablespoons water until a coarse paste forms, adding another tablespoon water if too thick to blend. Set aside.
Heat star anise in a dry 8-quart heavy pot over moderate heat, stirring occasionally, 10 minutes. Add coconut milk and peanuts and boil over moderately high heat, uncovered, stirring frequently, until very thick, about 5 minutes. Reduce heat to moderately low, then stir in curry paste and lemongrass paste and cook, stirring, 5 minutes. Add chicken, fish sauce, sugar, and salt and cook, uncovered, stirring frequently, 10 minutes.
Increase heat to moderate, then add peanut butter, squash, sweet potato, onion, bell peppers, yellow-fleshed potato, and carrot and cook, covered, stirring occasionally, 10 minutes.
Add remaining 2 1/2 cups water and bring to a boil over high heat, stirring occasionally. Reduce heat to moderately high and cook, uncovered, stirring occasionally, 8 minutes, or until vegetables are almost tender. Add beans and cook, stirring occasionally, until just tender, 6 to 8 minutes.
Discard star anise and serve curry over rice, with bread on the side.
Cooks' note: ·Curry can be made 1 day ahead, cooled, uncovered, then chilled, covered.