Combine sesame oil, garlic, ginger, salt, and pepper in a large glass bowl. Add the salmon to the mixture and let marinade for approximate 10 minutes.
While the salmon is marinating, place a collapsible steamer in a wide saucepan with tight-fitting lid and add enough cold water to almost come up to the bottom of the steamer. Cover the saucepan and bring the water to a rolling boil over high heat to create a head of steam.
Add the broccoli, snow peas, and carrots to the steamer. Cover the saucepan tightly again and lower the heat so the water simmers to maintain steam. Steam the vegetables until they are done to your liking. Using a potholder, grab the central stem and remove the steamer. Transfer the vegetables to a dish and set aside.
Remove salmon from marinade (reserve the marinade) and cut into bite size pieces. In a large skillet or stir-fry pan, heat olive oil and reserved marinade on medium-high heat. Add salmon, water chestnuts, and tamari. Cook until salmon is partly done, approximately 5 minutes.
Add steamed vegetables, spinach, and ¼ cup water to the skillet. Stir, cover, and cook for an additional 5 minutes.