In a small bowl, whisk together the soy sauce (or tamari/coconut aminos), lime juice, water, and cornstarch until smooth; set aside.
Season the sliced chicken lightly with salt.
Heat a large wok or large skillet over high heat until very hot, then add half of the oil.
Add the chicken and stir-fry, stirring occasionally, until cooked through and browned, about 3–4 minutes. Use a slotted spoon to transfer the chicken to a plate and set aside.
Reduce the heat to medium and add the remaining oil to the wok. Add the minced garlic and grated ginger and stir for about 20 seconds until fragrant.
Add the sugar snap peas and sliced carrots and stir over medium-high heat until tender-crisp, about 3–4 minutes.
Return the chicken to the wok, pour in the soy-lime mixture, toss to coat, and cook for another 30 seconds to 1 minute until the sauce thickens and everything is heated through.
Serve immediately, topped with sliced scallions.