Heat oil in a large pan over medium-high heat and fry the gyros meat until cooked through, about 5 minutes.
In a separate saucepan, bring the white wine to a boil over medium-high heat. Add the meat broth and return to a boil.
Reduce the heat to medium, stir in the cream while stirring constantly, and bring back to a boil. Season with the Metaxa and simmer for 5 minutes until the sauce thickens and becomes creamy.
Preheat the oven to 175°C (347°F) and butter a baking dish.
Transfer the seared meat to the baking dish and pour the sauce over it. Sprinkle with the grated Emmentaler.