Preheat oven to 350 degrees F (175 degrees C). Grease a baking sheet.
Brush 5 sheets phyllo dough with butter and stack sheet on top of each other. Repeat with remaining sheets.
Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add zucchini, broccoli, and string beans, cover, and steam until tender, 2 to 6 minutes.
Toss vegetables, feta cheese, and thyme in a large bowl; season with salt and pepper. Spread vegetable mixture over evenly over the stacks of phyllo dough. Roll phyllo dough, beginning widthwise, over the vegetables. Transfer both rolls to the baking sheet and brush with egg.
Bake in the preheated oven until brown and crisp, 30 to 40 minutes. Let vegetable rolls to rest for 5 minutes before slicing.