Preheat oven to 425°F with oven rack in lower-middle position.

Slit tenderloin lengthwise almost, but not quite, all the way through, to form a long pocket. Sprinkle tenderloin with 1/2 teaspoon salt, 1/2 teaspoon pepper, and 1/2 teaspoon thyme. Mix 2 cups blue cheese, bread crumbs, 1/4 teaspoon pepper, and remaining 1/4 teaspoon thyme in a small bowl. Stuff beef with blue cheese mixture; tie tenderloin crosswise at 1 1/2 inch intervals with butcher's twine. Coat with 1 tablespoon oil and sprinkle generously with salt and pepper.

Heat a large, heavy roasting pan over two burners on medium-high heat. When wisps of smoke begin rising from the pan, turn on exhaust fan and sear roast until well browned, about 2 minutes on each side. Transfer to a baking sheet. Reduce heat under roasting pan to low and add port, scraping pan to loosen brown bits. Pour port into a medium saucepan along with 1 cup cicken broth. Reduce to 1 cup; whisk in remaining 1/2 cup blue cheese; set aside.

Meanwhile, add remaining 2 tablespoons oil to pan. Add onion and cook, stirring until tender and well browned, 4 - 5 minutes. Add barley and cook, stirring constantly, until lightly toasted, 1 to 2 minutes. Stir in dried mushrooms, then white wine, and simmer until nearly evaporated. Add remaining 2 quarts chicken broth and bring to a simmer. Set a wire rack in the roasting pan and set roast on rack. Set pan in oven and cook until a meat thermometer inserted into thickest portion of roast registers 120 degrees for medium-rare, or 125 degrees for medium, 30 to 35 minutes. Remove roast from rack, and rack from pan. Let roast rest uncovered; return barley to oven and continue to cook until liquid evaporates and barley is tender with a little chewiness, about 20 minutes longer. Remove from oven and stir in spinach and Parmesan, stirring until spinach wilts.

Return port sauce to a simmer and whisk in constarch mixture to thicken slightly if necessary. Transfer warm barley to a serving bowl. Cut tenderloin into 1-inch thick slices and serve with port sauce.