Cut the broccoli tops into florets. Peel the stems of the broccoli and cut into 1" cubes. Core and cut the cauliflower into florets. Steam broccoli and cauliflower together in a steamer basket until crisp-tender. Do not overcook!
Toss the chicken in the soy sauce and then the flour to coat evenly.
Heat 1 tablespoon of oil in a wok or large skillet over high heat and stir fry the chicken breast until cooked through. Remove the chicken from the wok. Add another tablespoon of oil and stir fry the garlic, onions, and ginger root for 30 seconds and then stir in the curry paste and lime leaves. Return the chicken to the pan and combine with the curry mixture. Add in the coconut milk, fish sauce, soy sauce, and sugar. Stir until bubbling and add in the broccoli and cauliflower and red pepper flakes. Simmer and toss for 2 minutes to combine flavor.
Serve over basmati rice and garnish with cilantro.
