Bring broth to simmer in medium saucepan over medium-low heat then whisk in curry paste and sugar until dissolved. Add chicken and potato, bring to simmer, and cook until tender, 8 to 10 minutes.
Stir in bell pepper and snap peas and cook until just tender, 3 to 5 minutes.
Off heat, stir in coconut milk, basil, fish sauce, and lime juice and season with salt and pepper to taste. Serve.
Curry can be refrigerated for up to 3 days.