It's critical the vegetables are dry to the touch before you add them to the pan. Excess moisture will steam them, turning the dish into a soggy braise.
In a small bowl combine peanut butter, brown sugar, 2 Tbsp. of the soy sauce, lime juice, and crushed red pepper. Stir to combine. In another small bowl combine sherry and remaining 1 Tbsp. soy sauce; set aside.
Heat a 14-inch flat-bottomed wok or 12-inch stainless-steel skillet over high heat until a bead of water vaporizes within 2 seconds. Swirl in the 2 Tbsp. peanut oil. Add ginger; stir-fry 10 seconds or until fragrant. Add carrots; stir-fry 2 minutes. Add beans and cook 2 minutes or more or until beans are bright green. Swirl in sherry mixture and stir-fry 2 minutes more or until vegetables are crisp-tender and vegetables have absorbed most of the liquid. Add cabbage; cook 30 seconds or until slightly wilted.
To serve, drizzle with peanut butter mixture. Top with chopped eggs.