FOR THE STUFFING: Heat oil in medium saucepan over medium heat until shimmering. Add shallot and cook, stirring occasionally, until softened and golden brown, 2 to 3 minutes. Meanwhile, combine dried figs and port in microwave-safe bowl; cover with plastic and microwave on high until simmering, about 2 minutes. Set aside until needed.
When shallot is softened add dried fig/port mixture, thyme, salt, and pepper; continue to cook, stirring occasionally, until mixture is thick, 1 to 2 minutes. Remove pan from heat and stir in balsamic vinegar. Transfer dried fig mixture to plate. Set aside and cool to room temperature.
FOR THE ROAST: Following illustrations 1 and 2, butterfly roast. Season cut side of roast liberally with salt and pepper. Following illustration 3, spread cooled stuffing mixture in an even layer over interior of roast, leaving 1/2-inch border on all sides. Sprinkle pecans in an even layer on top of stuffing. Following illustrations 4 and 5, roll and tie beef roast.
In small bowl, stir together 1 tablespoon olive oil, 1 1/2 teaspoons salt, and 1 1/2 teaspoons pepper. Rub exterior of roast with oil mixture. Let roast stand at room temperature for 1 hour.
Adjust oven rack to middle position; heat oven to 450 degrees. Heat remaining 2 tablespoons olive oil in 12-inch skillet over medium-high heat until smoking. Add beef to pan and cook until well browned on all sides, 8 to 10 minutes total. Transfer beef to rimmed baking sheet and place in oven. Roast until instant-read thermometer inserted into thickest part of roast registers 120 degrees for rare, 16 to 18 minutes, or 125 degrees for medium-rare, 20 to 22 minutes.
FOR THE BUTTER: While meat roasts, stir all ingredients together in a small bowl until combined. Transfer tenderloin to cutting board; spread half of butter evenly over top of roast. Loosely tent roast with foil; let rest for 15 minutes. Slice roast between pieces of twine into thick slices. Remove twine from individual slices and serve with remaining butter passed separately.