Heat oil in a saucepan over medium-high heat. Add garlic; cook until browned, about 2 minutes. Transfer garlic to a bowl. Saute onion and achiote powder until onion is translucent, about 5 minutes. Add chicken, soy sauce, and fish sauce. Stir in bouillon cube until dissolved. Add 1 cup water; bring to a boil.
Soak vermicelli in a bowl of hot water, 2 to 3 minutes. Drain. Cut noodles to a desired length.
Drain mushrooms. Cut into 1-inch pieces.
Toss cabbage, bok choy, and carrot into the boiling water in the pan. Add the vermicelli, mushrooms, pepper, and chili powder. Cook, mixing well, until pancit is slightly dried, about 5 minutes. Arrange pancit on a platter. Garnish with the browned garlic and sliced eggs.