Heat 1 tablespoon olive oil in a large skillet over medium heat until shimmering.
Add the chicken pieces in an even layer, season with salt and pepper, and cook until lightly browned and cooked through, about 4 to 5 minutes, stirring as needed. Transfer chicken to a plate and set aside.
Add remaining tablespoon of olive oil to the hot skillet, then add broccoli, red bell pepper, sugar snap peas, and carrots. Sauté, stirring frequently, until vegetables are tender-crisp, about 4 to 5 minutes.
While vegetables cook, whisk together tamari (or soy sauce), peanut butter, honey, sesame oil, and grated ginger in a small bowl. Whisk in 2 to 3 tablespoons water or broth until sauce reaches your desired consistency.
Return the cooked chicken to the skillet, add the minced garlic and cashews, then pour the peanut sauce over everything. Stir to combine and bring the sauce to a gentle boil.
Reduce heat and simmer briefly until the sauce thickens slightly and coats the chicken and vegetables, about 1 to 2 minutes.
Sprinkle with chopped green onions, toss once more, then serve.