Combine green curry paste, garlic, ginger paste and lemongrass paste in a small bowl. Stir to combine.
If using homemade green curry paste, omit garlic, ginger and lemongrass.
Heat oil in a large Dutch oven over medium high heat. Turn on the exhaust over your stove top. Add curry paste mixture and cook for 2 to 3 minutes until it mostly dries out. DO NOT breathe in the fumes!
Add chicken broth and coconut milk; mix to dissolve the paste. Add fish sauce, sugar, and lime leaves. Mix and bring to a simmer.
Add chicken, stir and then lower the heat to medium so that it is bubbling gently. Cook for 7 minutes.
Add cauliflower and cook 2 minutes more. Add zucchini and mushrooms. Cook until just barely tender, about 3 minutes. Taste the sauce. Add fish sauce for more saltiness or sugar for more sweetness. Add snow peas and cook 2 minutes until a bit softened.
Remove from heat. Do not keep simmering or the sauce will darken. Stir in basil leaves and lime juice.
Serve curry over jasmine rice and garnished with cilantro.