Heat olive oil in a large skillet over medium heat. Add shallots and sugar; cook for 2 minutes, stirring constantly. Sprinkle with water; cover, reduce heat to medium-low, and cook for 10 minutes, stirring occasionally until shallots are soft. Add mushrooms; cook, uncovered, over medium-high heat 20 minutes or until liquid evaporates, stirring frequently. Add Marsala; cook 1 minute. Remove from heat, and cool. Stir in sour cream, parsley, salt, thyme, and pepper.
Place 1 phyllo sheet on a large cutting board or work surface (cover remaining dough to keep from drying), and lightly coat with cooking spray. Sprinkle with about 2 teaspoons breadcrumbs. Repeat the layers with 3 phyllo sheets, cooking spray, and breadcrumbs, ending with the phyllo.
Spoon 1 3/4 cups mushroom mixture along 1 long edge of phyllo, leaving a 1-inch border.
Starting at the long edge with the 1-inch border, roll up jelly roll fashion.
Place strudel, seam side down, on a baking sheet coated with cooking spray. Tuck ends under. Repeat the procedure with the remaining phyllo sheets, cooking spray, breadcrumbs, and mushroom mixture. Brush strudels with butter.
Bake strudels at 400° for 20 minutes. Let stand for 5 minutes. Cut each strudel into 4 slices.