In a small saucepan combine sugar and 1/2 cup water. In the microwave or stove-top, heat another 1/3 cup of water; set aside.
Heat saucepan with sugar over medium heat and stir until sugar is dissolved. Then, stop stirring and allow mixture to boil, untouched, for about 10-12 minutes, until mixture turns a deep golden brown. Watch carefully so as not to burn.
Turn off the heat and pour in the additional hot water. Do this very quickly and carefully so that you do not burn yourself. Mixture will sputter and bubble up a bit and then settle down.
Stir until caramel becomes liquidy. Set mixture aside to cool.
Meanwhile prepare a second bowl with shallots, soy sauce, fish sauce and olive oil. When caramel sauce is cool to the touch, add the caramel sauce to the shallot mixture.
In marinating dish (or ziplock bag) add chicken and pour marinade over the chicken.
Marinate for at least 1 hour and up to overnight.
To prepare the noodles:
Bring a large pot of water to a boil. Add vermicelli and boil per package instructions; about 5 minutes. Drain into a colander and run cold water over the noodles until they are cool to the touch. Set aside to drain while you prepare your remaining ingredients.
To prepare the Nuoc Cham:
Combine water, fish sauce, lime juice, sugar, vinegar, garlic, thai chili, shredded cucumber and shredded carrot. Stir until the sugar is completely dissolved. Set aside.
To grill the Chicken:
Brush a grill pan with vegetable oil and bring to high heat.
Remove chicken from marinade and blot with paper towel to remove excess moisture.
Grill for about 3-4 minutes per side or until chicken is completely cooked through.
Let rest for 5 minutes before slicing.
Prepare the noodle bowls by distributing a handful of lettuce, 1/4 portion of noodles, and top with shredded carrot, cucumber, chopped peanuts, mint and cilantro to taste.
Top salads with sliced grilled chicken.
Separate the Nuoc Cham into 4 portions for serving. Pour sauce over salad and mix immediately prior to eating.