In a saucepan of boiling salted water, cook sweet potato until tender, about 12 minute Drain and set aside.
Meanwhile, heat oil in a skillet over medium-high heat.
Add mushrooms, onions and thyme and saute until onions have begun to caramelize, about 10 minute Add garlic and cook for 2 minute Stir in sweet potatoes, arugula, salt and pepper.
Set aside to cool.
Using 3 sheets of phyllo for each strudel, brush and stack 3 sheets with some of the melted butter.
Spoon half the mushroom filling in the centre of phyllo, leaving 3-inches (7.5 cm) of pastry empty on each end.
Sprinkle half the cheese over filling.
Fold long edges over filling to create a log.
Fold ends underneath to seal package.
Place on a parchment paper-lined baking sheet.
Repeat with remaining phyllo, filling and cheese.
Brush tops with remaining butter.
Bake until golden and crisp, about 20 minute Cut each strudel into 4 pieces.