Stir together 1 Tablespoon of the hoisin, 1 Tablespoon of the soy sauce, and the cornstarch until the cornstarch is dissolved in a medium bowl. Add the ground pork (or turkey) and mix until the sauce is evenly distributed throughout. Set aside.
Heat 1 Tablespoon of the oil in a 12-inch skillet over medium-high heat. Add the sliced mushrooms and cook until browned (about 5 minutes), stirring occasionally. Stir in the ginger and garlic and cook until fragrant, about 30 seconds.
Stir in the chicken broth and the remaining 3 Tablespoons of hoisin sauce and 3 Tablespoons of soy sauce. Place the noodles on top of the broth and cover. When the broth comes to a boil, reduce the heat to medium and continue to cook until the bottoms of the noodles have softened, 3-5 minutes. Using tongs, flip the noodles and sprinkle the broccoli over the top of the noodles. Cover and cook for another 3-5 minutes, stirring halfway through.
Transfer the noodle mixture to a large bowl or platter and cover with foil. Add the remaining 1 Tablespoon of oil to the skillet and heat over medium-high heat. Add the ground pork mixture and cook until browned on the bottom (about 2 minutes), making sure to break up any large clumps. Stir the meat and cook until browned through about 2 more minutes.
Distribute the pork over the top of the noodles, and serve immediately.