Preheat oven to 350°F (180°C). Heat half the oil in a large skillet over medium-high heat; add shallots and sauté for 1 min. Add all of the mushrooms and sauté for 10 min. or until most of the moisture has evaporated. Mix in asparagus and sauté until slightly tender. Cool slightly, then sprinkle with parsley, lemon zest, lemon juice, pepper and 3 tbsp (45 mL) Parmesan cheese. Mix to combine and set aside.
Working on a clean countertop, brush the first sheet of phyllo with olive oil and sprinkle with 1/4 tsp (1 mL) thyme. Repeat with remaining phyllo sheets, stacking one on top of the other with the longer sides parallel to the counter’s edge.
Place filling along the longer edge of phyllo and roll into a log, tucking in the sides to prevent mixture from falling out. Brush strudel with olive oil and top with remaining cheese. Place on parchment-paper-lined baking sheet with seam on the bottom and cut three slits widthwise into top of strudel. Bake for about 17 min. or until golden brown. Cool slightly. Slice where strudel is slit to make four equal portions, then serve.