Whisk the tamari, fish sauce, and coconut sugar in a small bowl until the sugar dissolves.
Mince the garlic, shallot, and bird's eye chilies together into a fine paste using a knife or mortar and pestle.
Heat half of the avocado oil in a large wok or skillet over high heat until it begins to shimmer.
Add the garlic-chili paste to the hot oil and sauté for 30 seconds until intensely fragrant and golden.
Add the ground chicken to the pan, breaking it into small crumbles with a spatula, and cook until browned and no longer pink.
Toss in the sliced red bell pepper and trimmed green beans, stir-frying for 2-3 minutes until the vegetables are tender-crisp.
Pour the prepared sauce over the chicken and vegetables, tossing constantly to glaze every piece evenly.
Remove the pan from the heat and stir in the fresh Thai basil leaves until they are just wilted.
In a separate small non-stick skillet, fry the egg in the remaining avocado oil until the edges are crispy and the yolk remains runny.
Portion the warm quinoa into a bowl, top with the spicy chicken mixture, and finish with the fried egg and a fresh squeeze of lime juice.