Heat olive oil in a large pot over medium heat. Add the diced chicken and cook until browned. Remove from the pot and set aside.
In the same pot, sauté onion and garlic until fragrant, about 2 minutes.
Add carrots and celery, and cook for another 5 minutes, stirring occasionally.
Pour in the chicken broth and bring to a boil. Reduce heat to simmer and add the broccoli and snap peas.
Stir in soy sauce, grated ginger, salt, and pepper. Let the soup simmer for 10 minutes.
Return the cooked chicken to the pot and simmer for an additional 5 minutes.
Serve hot, garnished with green onions and sesame seeds if desired.