Heat a large skillet over medium heat. Coat the pan with nonstick cooking spray. Add the beaten eggs and stir with a spatula almost constantly for 3 minutes, until scrambled, fluffy, and cooked through. Remove to a small plate. Wipe out the skillet.
Add sesame oil to the skillet and increase the heat to medium-high.
Add the carrots, edamame, scallions, mushrooms, ginger, and garlic. Saute, stirring frequently, for about 5 minutes.
Add the rice. The pan of veggies and rice will seem dry but you want the rice to brown and become crispy. Saute the whole mixture for about 2 minutes.
In a small bowl, whisk together the soy sauce and brown sugar until the sugar has dissolved. Pour into the pan. Stir the mixture to coat everything evenly.
Drizzle everything with additional sesame oil (about 2 teaspoons- important for flavor)
Add the scrambled eggs back to the pan and mix well. Serve.