Pulse all marinade ingredients in a mini food processor. It'll take about 10 pulses to resemble a slightly coarse puree. Transfer mixture to a large bowl.
Cut chicken into 1-inch cubes and place into marinade bowl. Using your hands, coat and massage the chicken in the marinade then thread onto skewers. If using wooden skewers, make sure to soak them in water for 5-10 minutes prior. Let chicken marinade at room temperature for 30 minutes to an hour.
In the meantime, cook the noodles per package instructions and prepare all the toppings and Nuoc Cham.
When chicken is done marinading, turn outdoor grill on high. Cook chicken on the grill, turning every so often until chicken is cooked through (165 °F (74 °C) internal temperature).
Assemble your noodle bowl with a bed of noodles, veggies/toppings, and skewered chicken on top.