Fill a large Dutch oven about 2/3 full with water. Bring water to 130 degrees F (54 degrees C) over medium heat. Attach a candy or oil thermometer to the side of the Dutch oven in order to monitor and maintain the water temperature.

Generously season steaks with salt and pepper. Heat 1 teaspoon vegetable oil in a large skillet over high heat. Add steaks and cook until just browned, about 1 minute per side. Transfer steaks to a resealable plastic bag, squeeze out all air, and seal.

Place the sealed plastic bag into the Dutch oven, adjusting the heat as necessary to maintain the water temperature at 130 degrees F (54 degrees C). Cook for 2 1/2 hours, moving the bag around occasionally and maintaining the water temperature.

While the steaks are cooking, heat 1 tablespoon butter in the same skillet over medium-low heat. Add mushrooms and cook, stirring occasionally, until soft, about 5 minutes. Transfer to a plate and set aside.

Remove steaks from the plastic bag (reserving any accumulated juices) and pat dry. Generously season again with salt and pepper. Heat remaining 1 teaspoon vegetable oil in the skillet over high heat. Place steaks in the skillet and cook to form a crust, about 2 minutes per side. Transfer steaks to a plate.

Stir garlic and mushrooms into the skillet, reduce the heat to medium, and cook until mushrooms are reheated, about 2 minutes. Stir in reserved steak juices, 2 tablespoons water, and balsamic vinegar. Remove from the heat.

Stir in remaining 1 tablespoon butter until completely melted and incorporated. Season with salt and pepper to taste. Spoon sauce over steaks and serve.