In a large bowl, toss the chicken with 1 Tbs. of the soy sauce, 1-1/2 tsp. of the sesame oil, and 1 tsp. of the rice vinegar.
In a 1-cup liquid measuring cup or another bowl, combine 1/4 cup water with the ketchup and the remaining 1 Tbs. soy sauce, 1 Tbs. vinegar, and 1-1/2 tsp. sesame oil.
Trim the scallions and separate the dark-green tops from the light-green and white bottoms. Slice the tops into 2-inch pieces and the bottoms into thin rounds. Combine both in a medium bowl with the snow peas and ginger.
Heat 1-1/2 Tbs. of the oil in a 12-inch nonstick skillet or large stir-fry pan over medium-high heat until shimmering hot. Add the chicken and cook, stirring occasionally, until it loses most of its raw appearance, 1 to 2 minutes. Transfer to a large plate.
Add the remaining 1-1/2 Tbs. oil and the scallions, snow peas, and ginger, and cook, stirring occasionally, until the ginger and scallions start to brown, about 2 minutes.
Return the chicken to the pan and add the ketchup mixture and half of the sesame seeds. Cook, stirring, until the chicken is cooked through and the snow peas are crisp-tender, 2 to 3 minutes. Transfer to a platter, sprinkle with the remaining sesame seeds, and serve.