Stir together chicken broth, tamari, fish sauce, sugar, and cornstarch/GF flour in a small dish then set aside. Whisk eggs together in another small dish then set aside.
Heat sesame oil in a large wok or 12" skillet over medium-high heat. Add chicken, season with salt and pepper, then stir fry until cooked through and remove to a plate.
Heat 2 teaspoons cooking oil in skillet then add shallots and peppers. Season with salt and pepper then stir fry until crisp-tender, 5 minutes. Add garlic then stir fry for 30 more seconds. Remove vegetables to plate with chicken.
Heat remaining 2 teaspoons cooking oil in skillet then add cabbage. Season with pepper only then stir fry until just barely crisp tender, 3-4 minutes (it will continue to cook and soften in the sauce.) Add eggs then cook until set, 1-2 minutes. Add the tamari sauce mixture then bring to a bubble and boil until slightly thickened, 1-2 minutes.
Remove skillet from heat then stir in chopped cilantro. Divide Pad Thai between plates then top with chopped peanuts and drizzle with fresh lime juice.