To make the Béarnaise butter:
Melt the 2 tablespoons butter in a small sauté pan over medium heat and add the shallots. Sweat until the shallots are soft, about 2 minutes. Add the wine and vinegar and reduce until the liquid has evaporated. Remove from the stove and add shallot mixture to a mixing bowl to cool. When cooled, stir in the softened butter, tarragon, and black pepper. Mix well and chill until ready to use.
For the grilled steaks and onions:
Heat your gas grill to medium-high or start you charcoal grill. Remove the steaks from the fridge and set aside at room temperature for up to ½ hour for better grilling. Remove the béarnaise butter from the fridge and set aside at room temperature to soften.
Rub the onion slices carefully with the oil, sprinkle with the thyme, and season with the salt and pepper. When the grill is hot, or before your charcoal is white hot, grill the onion slices, about 4 minutes for each side. The onions should be soft and lightly charred. Set aside while you grill the steaks. Turn the gas grill up to high. Season the steaks generously with salt and pepper. When hot, or your when your charcoal is white hot, grill the steaks to desired doneness. Remove from the grill and let rest for five minutes before slicing. Heat butter in microwave for a few seconds just to bring it to a spoonable consistency. It is unnecessary to microwave until the butter is hot as the heat from the steak will continue to make it melt.
When ready to serve, slice the steaks into 1/3 inch slices. (You can leave the steaks whole if you wish.) Arrange some of the grilled onions on a plate, top with the sliced steak and spoon over some of the béarnaise butter.