Make the marinade: Mix Burgundy wine, oyster sauce, canola oil, soy sauce, garlic, and dried oregano in a medium saucepan. Bring to a boil, then remove from heat. Place in the refrigerator until chilled, about 1 hour.

Marinate steaks: Place steaks in a 9x13-inch baking dish and pour chilled marinade over them. Cover tightly with foil and refrigerate for at least 3 hours, or overnight if possible.

Make the butter: Cream softened butter and Burgundy wine in a bowl using an electric mixer. Mix in shallots, green onions, and white pepper by hand. Cover tightly and refrigerate until needed.

Preheat grill and oven: Preheat an outdoor grill to high heat and lightly oil the grate. Preheat the oven to 200°F (95°C).

Prepare steaks: Remove steaks from marinade and shake off excess. Discard remaining marinade.

Grill steaks: Grill on the preheated grill until hot and slightly pink in the center, approximately 5 to 7 minutes per side (medium level). Internal temperature should reach 140°F (60°C).