Place cast-iron skillet on stovetop over medium heat. Add 2 tablespoons ghee and let melt completely, swirling and tilting pan as needed to distribute ghee across entire surface.
When ghee has melted, place 4 ½-pound beef burger patties in skillet. Cook patties 3 to 6 minutes, then flip patties over and cook 3 to 6 minutes on other side, until burger patties are fully cooked-through.
Transfer fully-cooked burger patties to large plate and set aside.
Return skillet to stovetop and increase heat to medium-high. Add 14 slices white onion to skillet. Sauté onions, stirring occasionally, approximately 5 minutes or until onions are translucent.
When onions are translucent, add 12 ounces sliced baby bella mushrooms and stir to incorporate. Sauté mushrooms, stirring occasionally, approximately 3 minutes or until mushrooms begin to soften.
Add 4 cloves garlic and additional ghee if needed. Sauté garlic, stirring constantly, until garlic is fragrant, approximately 30 to 60 seconds.
When garlic is fragrant, add 2 teaspoons anchovy paste (if using) and 2 tablespoons tapioca starch. Whisk anchovy paste and tapioca starch into existing liquid in skillet until mixture is smooth.
Slowly pour 1 14.5-ounce can compliant beef broth into skillet in small increments, whisking constantly to incorporate beef broth fully before adding more.
Once all beef broth as been added, increase heat under skillet to medium-high. Bring liquid to boil. Once boiling, immediately reduce heat under skillet to low.
Remove skillet from heat and let cool slightly. Add ½ teaspoon garlic powder, ½ teaspoon onion powder, ½ teaspoon freshly ground black pepper, ½ teaspoon salt, and 1 cup unsweetened, full-fat coconut milk to skillet. Whisk ingredients together until fully combined.
Return skillet to low heat. Return fully-cooked burger patties to skillet and spoon gravy from skillet onto patties. Simmer burger patties in gravy 3 to 5 minutes or until patties are warmed-through and gravy has thickened slightly.
Portion salisbury steaks onto serving plates with mashed potatoes or mashed cauliflower. Spoon plenty of gravy onto steaks and serve immediately.