Remove roots, outer leaves, and tops from leeks, leaving 6 inches of each leek. Cut each diagonally into thirds, then diagonally in half to form 6 triangular pieces. Rinse under cold water; drain well.
Combine beef broth, tomato paste, salt, thyme, and pepper in a bowl, and stir with a whisk.
Melt butter in a large nonstick skillet over medium-high heat. Add leeks and mushrooms, and sauté 6 minutes or until the vegetables are lightly browned. Add broth mixture. Cover, reduce heat, and simmer 15 minutes or until leeks are tender. Uncover, and simmer 7 minutes or until liquid almost evaporates, stirring occasionally.