Heat the oil in a 2- to 3-quart saucepan over medium heat until a bit of curry paste just sizzles when added to the pan. Add all the curry paste and cook, pressing and stirring with a wooden spoon or heatproof spatula to soften the paste and mix it in with the oil, until fragrant, about 2 minutes.
Add the coconut milk and broth and bring to a simmer. Simmer, stirring often, for 5 minutes, allowing the flavors to develop.
Increase the heat to medium high and let the curry come to a strong boil. Add the shrimp, sugar snap peas, and half the lime leaves (if using), and stir well. Cook, stirring occasionally, until the shrimp curl and turn pink, about 2 minutes. Add the fish sauce, sugar, and salt and stir to combine. Remove from the heat.
Tear the basil leaves in half (or quarters if they are large), and stir them into the curry, along with the remaining lime leaves (if using). Let rest for 5 minutes to allow the flavors to develop.
Serve hot or warm with rice or noodles, garnished with the chile slices (if using).