Slice the beef across the grain 1/4 inch thick and then cut the slices into 1-1/2- to 2-inch-long pieces.
Trim and discard the stems from the shiitakes; slice the caps 1/4 inch thick (you should have 1-1/2 to 2 cups).
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.
Increase the heat to medium high, and add the beef. Spread it in an even single layer and cook undisturbed until it just begins to lose its pink color, about 1 minute. Turn the beef and continue cooking, stirring occasionally to coat it with the curry paste, until most of the beef no longer looks raw, 1 to 2 minutes. Stir the shiitakes into the beef.
Add the chicken broth and stir again. Add half the lime leaves (if using), and bring to a simmer. Simmer gently, stirring occasionally, until the shiitakes are tender and the beef is cooked through, about 5 minutes.
Add the edamame, stir well, and cook for about 1 minute, just to blanch them. 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. Season to taste with salt.
Serve hot or warm with rice or noodles, garnished with the chile slices (if using).
Serve the curry with lime wedges on the side to squeeze over all.