Heat half the oil in a large skillet over medium-high heat.
Saute shallots and mushrooms until most of the moisture has evaporated.
Mix in asparagus and saute until slightly tender.
Spinkle with parsley, lemon zest, lemon joice, pepper and 3 tbsp Parmesan cheese. Mix.
Brush first sheet of phylio with olive oil and spinkle with 1/4 tsp tyme. Repeat with the remaining phylio sheets, stacking one on top of the other with the longer sides parallel to the counter's edge.
Place filling along the longer edge o phylio and roll inot a log, ticking 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 o the bottom and cut three slits widhwise into the top of the strudel.
Bake for about 17 minutes or until golden brown.
Slice whre strudel is slit to make four equal portions, then serve.