Add olive oil, soy sauce, Mirin, brown sugar, and minced garlic to a large zip-lock bag. Cut steak into bite-sized pieces and add to marinade. Seal bag and shake until steak is fully coated; place in the refrigerator for at least 30 minutes.
While steak marinates, prepare the dressing for the salad: add olive oil, soy sauce, the juice of the lime, minced garlic, and grated ginger to a mixing bowl; whisk well and cover. Place in the refrigerator until ready to serve.
Heat a large non-stick skillet over medium. Dice onion and add to hot skillet along with marinated steak; discard the marinade. Saute until meat is browned and cooked through as desired.
Add spinach to a large mixing bowl. Halve tomatoes and thinly slice green onion; add both to the spinach and toss with the refrigerated dressing. Top with steak and onions.
Serve with orange wedges and sliced Italian bread on the side.