Cook quinoa according to package directions, omitting salt, about 14 min. Transfer to a large bowl.
Heat 2 tbsp of olive oil in a large skillet over medium heat; add mushrooms, onion, and garlic to the hot oil and season with salt, black pepper, and cayenne.
Cook and stir mushroom mixture until mushrooms have given up their juice and the juice has almost evaporated, about 10 minutes. Transfer cooked mushrooms to food processor and pulse. Add Woolwich Dairy Original Soft Fresh Goat Cheese Log, 2 beaten eggs, cooked quinoa, and breadcrumbs and pulse until mixture takes on a paste-like consistency.
Heat the same non-stick frying pan over medium heat. Firmly press mushroom & quinoa mixture into a 1/2-cup measuring cup. Turn and release into pan. Gently press to shape into a patty about 4 inches wide. Repeat, cooking 2 patties at a time. Cook until golden and warmed through, about 4 min per side.
Once patty is cooked, add 1 slice of Armstrong Havarti cheese to each patty.
Remove patty from skillet once cheese has slightly melted.