Place 1 sheet of the phyllo pastry on the work surface. Brush with some of the butter and sprinkle with some of the Parmesan. Top with another sheet; brush with more butter and sprinkle with more Parmesan. Top with third sheet and brush with more butter.
Cut the phyllo stack in half lengthwise. Repeat with the remaining 3 phyllo sheets, reserving some of the Parmesan.
Lay the asparagus spears along the long edge of each phyllo stack. Dot with some of the chèvre and sprinkle with green onion. Roll up.
Brush with butter and sprinkle the remaining Parmesan.
Place, seam side down, on a baking sheet and bake until golden.