In a small bowl, stir together the oyster sauce and 2 tablespoons soy sauce. Set aside. Keep a large bowl or plate and a large, flexible rubber spatula near the stove.
Heat 12-inch nonstick skillet over medium heat until hot, about 2 minutes. Add 1/2 tablespoon butter and swirl to coat the bottom of the pan. Add the eggs. Cook without stirring until they barely begin to set, about 20 seconds. With your spatula, scramble and break the eggs into little, bite-sized pieces. Continue to cook, stirring constantly now, until eggs are just cooked through but not yet browned, about 1 additional minute. Transfer eggs to the bowl and set aside. With a paper towel, carefully wipe the skillet clean.
Return the skillet to the heat, and increase the heat to high. Let the skillet warm until it is nice and hot, about 1 minute. Add 1 tablespoon of the canola oil, and swirl to coat. Add the chicken and the remaining 1 tablespoon soy sauce. Cook, stirring occasionally, until the chicken is fully cooked through, about 4 minutes. Remove to the bowl with the eggs.
Add the remaining 1 tablespoon oil to the skillet. Add the fresh vegetables and cook until they are crisp-tender, about 4 to 5 minutes.
Add the remaining 1 1/2 tablespoons butter and peas and carrots. Cook, stirring constantly, for 30 seconds.
Stir in the garlic and cook until fragrant, about 30 seconds (do not let the garlic burn!).
Add the brown rice and the oyster sauce mixture. Continue cooking, stirring constantly and breaking up any remaining rice clumps, until the mixture is heated through, about 2 minutes. Add the reserved eggs/chicken and green onions. Cook and stir until the mixture is completely heated through, about 1 minute more. Enjoy immediately with a sprinkle of red pepper flakes or dash of hot sauce and additional soy sauce as desired.
ABOUT OYSTER SAUCE: Oyster sauce is readily available in the Asian or international food aisle of most major grocery stores. It has a unique flavor and texture, and if you substitute something else for it, your fried rice won't taste the same. If you'd like to experiment with making the recipe without oyster sauce, you can simply omit it or swap in 1 tablespoon sesame oil to add flavor.
TO STORE: Refrigerate fried rice in an airtight storage container for up to 3 days.
TO REHEAT: Rewarm leftovers in a skillet on the stovetop over medium-low heat or in the microwave.
TO FREEZE: Freeze chicken fried rice in an airtight, freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating.