2 tablespoons butter
4 cups thinly sliced onion (4 large)
1 tablespoon snipped fresh thyme or 1 teaspoon dried thyme, crushed
1 teaspoon sugar
1 teaspoon salt
1/2 teaspoon freshly ground black pepper
1 pound beef tenderloin
1/8 teaspoon ground black pepper
4 ounces crumbled blue cheese
4 teaspoons butter
4 sourdough buns or kaiser rolls, split
For caramelized onion: In a large nonstick skillet, melt butter over medium-low heat. Add onion. Cook, covered, for 15 minutes, stirring occasionally. Uncover; stir in thyme, sugar, salt, and 1/2 teaspoon pepper. Cook, stirring occasionally, over medium-high heat for 14 to 16 minutes more or until onions are extremely tender and golden. Remove from heat; set aside.
Trim fat from meat. Cut meat crosswise into four slices. Place each slice between two pieces of plastic wrap. Use the flat-side of a meat mallet or a rolling pin to lightly pound meat to 1/2-inch thickness. Discard plastic wrap. Season beef with 1/8 teaspoon pepper.
For charcoal grill: Place meat on rack of uncovered grill directly over medium coals. Grill steak for 3 minutes, turning once halfway through. (For a gas grill: Preheat grill. Reduce heat to medium. Place meat on grill rack over heat. Cover; grill as directed.) Remove meat from grill. Top each steak evenly with the reserved caramelized onions and blue cheese. Return meat to grill. Grill 2 to 3 minutes more or until cheese begins to melt.
Meanwhile, heat 2 teaspoons butter on a griddle over medium heat. Lay two buns, cut sides down on griddle pan. Toast buns for 2 to 3 minutes or until golden brown. Repeat with remaining butter and buns.
Serve steaks on toasted buns.