1. In a large nonstick skillet, heat oil over medium-high heat. Add mushrooms and 1/4 teaspoon each salt and pepper, and cook, stirring, until tender and golden (4-6 minutes). Transfer mushrooms to a plate. Wipe out skillet; reserve.
2. With clean hands, form meat into 4 patties; sprinkle remaining 1/2 teaspoon pepper over both sides of burgers. Return skillet to medium-high heat; cook burgers until medium (5-6 minutes per side), or until a meat thermometer inserted in the center reads 160°.
3. Place 1/4 of the radicchio slices, then burgers, on bottom buns; divide mushrooms among burgers and top with other half of bun.