Whisk together the soy sauce, brown sugar, and cornstarch in a small bowl. Set aside.
Heat the oils in a large skillet or wok over medium-high heat. Add the garlic and ginger and cook until fragrant, about 30 seconds. Add the turkey and cook until crumbly and cooked through.
Add the cruciferous crunch, carrot, and most of the green onions (save some for garnish). Cook until tender, about 5 minutes.
Pour in the reserved sauce and cook until the mixture is thickened, about 2 minutes.
Divide the rice between serving bowls and top evenly with the egg roll mixture. Garnish with more green onions and wonton strips.