In a medium saucepan, mix together Burgundy wine, canola oil, soy sauce, oyster sauce, garlic & oregano. Bring to a boil, & then remove from heat. Place in the refrigerator for 1 hour, or until chilled.
Place filet mignon filets in a 9×13 inch baking dish, & pour the chilled marinade over them. Cover tightly with foil, & refrigerate for a minimum of 5 hours.
In a medium bowl, cream butter & 1 teaspoon of Burgundy wine with a hand mixer. Mix in shallots, green onions & white pepper by hand; cover tightly & refrigerate.
Preheat an outdoor grill for high heat & lightly oil grate. Preheat oven to 200 degrees F (95 degrees C).
Grill marinated fillets to the desired doneness, turning once. Place fillets in a clean 9×13 inch baking dish. Dollop with the Burgundy butter mixture, & place in the preheated oven for a minute, or until butter is melted.