1: Heat the 2 tablespoons of butter in a small saucepan over medium-low heat. Saute the shallots until they are caramelized and very soft, 15 to 20 minutes, stirring often. Remove from the heat and let cool.

2: In a small mixing bowl, combine the 8 tablespoons of butter, the shallots, Parmigiano-Reggiano, tarragon, cognac, and lemon juice and stir to combine. Add salt and pepper to taste. Place a 12-inch square of parchment paper, wax paper, or plastic wrap on your work surface. Mound the bearnaise butter in the center, then roll the butter into a cylinder about 1 1/4 inches in diameter. Twist the ends so the butter maintains its shape, then refrigerate or freeze until firm, about 2 hours. (The butter will keep in the refrigerator for up to 5 days, or can be frozen for 3 months.)

3: When ready to cook the steaks or up to 45 minutes in advance, brush the steaks on both sides with olive oil. Season evenly with coarse salt and cracked black pepper.

4: Build a wood fire in your grill and heat one side of it to medium-high. Leave the opposite side fire-free. Brush or scrape the grill grate clean and oil it.

5: Arrange the steaks on the grill grate and sear on both sides, turning them at a 90 degree angle to obtain a handsome crosshatch of grill marks. Slide the steaks to the fire-free zone to continue cooking.

6: When the steaks are cooked to your liking—125 to 135 degrees is rare to medium-rare—transfer the steaks to plates or a platter. Top each with a disk of béarnaise butter, allowing it to melt from the residual heat. Serve immediately.