1. Preheat oven to 400°.
2. Pierce each beet with a fork; wrap in foil. Bake at 400° for 1 hour and 15 minutes or until tender; cool. Peel red beet; cut into 8 slices. Place in bowl with 2 teaspoons olive oil, 1/8 teaspoon salt, and 1/4 teaspoon pepper. Toss gently. Repeat the procedure with golden beet and remaining oil, salt, and pepper.
3. Reduce oven temperature to 375°.
4. Heat butter in a large skillet over medium-high heat. Add mushrooms and the next 4 ingredients (through shallot); sauté 4 minutes or until tender, stirring frequently. Add nuts, broth, and wine; cook 1 minute or until liquid evaporates. Transfer mushroom mixture to a medium bowl; cool. Stir in cheese.
5. Place 1 phyllo sheet on a large cutting board (cover the remaining dough to prevent drying); coat phyllo sheet with cooking spray. Top with another phyllo sheet; coat with cooking spray. Gently press phyllo layers together. Arrange 2 slices red beet in center of 1 short edge of phyllo, leaving a 4-inch border. Top with 1/2 cup mushroom mixture; arrange 2 slices golden beet over top of mushroom mixture. Fold over long edges to cover.
6. Starting at the short edge with 4-inch border, roll up jelly-roll fashion. Place packet seam side down on a baking sheet coated with cooking spray; brush with egg. Repeat with remaining phyllo, red beets, mushroom mixture, golden beets, and egg. Bake at 375° for 25 minutes or until golden.