Take the rice out of the refrigerator, give it a good shake (in a covered bowl), and let it set at room temperature.
Warm soy sauce, water, mirin, sugar, garlic, and butter until diluted.
Debone the duck and chop all of the meat into thin slices/chunks. If you have skin still on your duck, you can decide to discard it or keep a little in to flavor the meat a bit more.
Heat olive oil in the pot on high heat (we'll be cooking everything on high heat until we mix everything back in) and cook the duck until it's done. If you're not familiar with duck, just make sure to cook like you would beef, for this recipe. Remove the duck, put in a separate container, and set aside.
Add all of the veggies and cook on high for about 8 minutes. 5 minutes seems to be too fast and doesn't get the veggies cooked enough and 10 minutes seems to be just about too long. The best tip I can give you is to taste test and make sure the veggies are still crisp, but done. Also, know that they'll continue cooking after you take them off of the oven. Remove the veggies, put in a separate container, and set aside.
Add about a tablespoon of soy sauce to the bottom of the hot pan. It'll start sizzling. You want it to cook a bit to deepen the flavor, but not necessarily burn, so keep stirring at this point. After about 2 minutes, add in your rice, soy mix (step 2; yet, add a bit at a time to your taste), and mix the ingredients together lightly. We don't want to destroy the rice here; we want to blend the flavors. A wooden spoon or rice fork/spoon works well here. Since we're still cooking on high heat, you need to keep mixing... gently.
Take the pot off of the heat and stir in the scrambled eggs. A lot of people make a well or mix the egg separately from the fried rice recipe, but I just mix it in with the hot rice.
Notes
*The amount of veggies are just a suggestion. Add more or less, depending on your taste. Also, try to vary it up between black pepper and white pepper to find what you prefer.