Brush the cut sides of the squash with 1 tablespoon of the peanut oil.
Place the squash, cut sides down, on a baking sheet and roast for 30 - 40 minutes. The squash is cooked when a knife easily pierces through the skin and flesh. Let cool, use a fork to shred the flesh into spaghetti like strands, and set aside.
In a small bowl, stir together the fish sauce, tamari, and red pepper flakes.
Heat 1 tablespoon of the oil in a large wok or large cast-iron pan over medium-high heat. Add the egg and cook until scrambled, 30-60 seconds, breaking it up. Transfer to a plate and reserve.
Pour another tablespoon of the oil into the wok and stir-fry the garlic until aromatic. Season the chicken with a little salt and add it to the pan. Stir-fry the chicken until golden brown, about 4 minutes. Push the chicken to the side and add the shrimp in the center of the pan. Stir frequently until the shrimp are pink and firm, and about 3 minutes.
Toss in the carrots and stir-fry them 1 minute. Transfer the contents of the pan to a platter.
Add the remaining 1 tablespoon oil to the wok, spread the squash strands out in the pan and cook for 1 minute without stirring. Flip the pile of strands over and brown them for 2 minutes on the other side.
Pour the sauce into the wok and add the chicken-shrimp mixture, egg, bean sprouts, and scallions. Gently toss to heat through, and squeeze the juice of one lime half over all. Garnish with the peanuts and serve with the other half lime available for table-side squeezing. Any leftovers can be refrigerated for up to 4 days.