Cook asparagus spears for one minute in a large amount of rapidly boiling water. Carefully drain, and then plunge spears into ice cold water. Drain again and pat dry.
Brush one sheet of filo lightly with butter and then top with the other sheet, also brushing it lightly with butter. Cut filo into 12 strips. Place 2 to 3 (depending on size) asparagus spears at the bottom of each strip, then tightly roll and place on a baking sheet. Brush with butter and bake in 400 degrees F oven for 10 minutes, or until filo is puffed and lightly browned. Serve warm or at room temperature, with mayonnaise alongside.
*Note: Unused filo can be tightly wrapped and stored in the fridge for a few weeks. Pesto can be bought in the deli.