Heat a tablespoon of vegetable oil in a wok over medium-high heat. As the oil is heating (before it gets to full heat), add curry paste, and half the broth, stirring until smooth. (Reserve the remaining broth to add to the wok if mixture becomes dry and curry begins to sizzle on the bottom of the wok.) Continue heating and stirring until the curry becomes fragrant and the wok has reached full heat, 1-2 minutes.
Add chicken and stir fry until just cooked through, about 4 minutes.
Add green beans, peppercorns, kaffir lime leaves, shallots, palm sugar, fish sauce and stir quickly to combine. Continue cooking for 2 minutes. If mixture becomes dry, add additional broth, ¼ cup at a time. (Mixture should be smooth but should thick enough to coat the other ingredients.)
Add basil and cook just until basil begins to wilt, about 1 minute. Remove from heat and serve immediately.