Place chicken wings on a roasting pan or casserole dish and roast for 30-35 minutes
Reduce the heat to 375°
Add the carrots and stir to combine. Roast for 20 more minutes.
Transfer the chicken and carrots to a large stockpot
Place the now-empty roasting pan or casserole dish on a stove-top over high heat
Add 2 cups of water, stirring and scraping vigorously with a heatproof or metal spoon. You want to get all the bits and pieces into the liquid. Once it begins to boil, add it to the stockpot.
Add the garlic, ginger, mushrooms, and remaining water to the stockpot. Bring this to a simmer over high heat.
Reduce the heat to as low as your stove allows
Add the soy sauce and allow it to continue cooking for 3 to 3 ½ hours. Occasionally skim the fat and scum into the trash.
Strain the broth to remove all the solid bits, discarding the solids
You can put the broth into the fridge for up to 3 days, until read to use. When ready to use, skim off any fat or scum off the top into the trash and bring the broth to a simmer.