Stir together ricotta, parsley, milk, thyme, garlic, and 3/8 teaspoon salt.
Place 1 phyllo sheet on a baking sheet lined with parchment paper. (Cover remaining dough to keep it from drying out.) Lightly brush phyllo sheet with oil, and top with another phyllo sheet. Repeat layers with remaining phyllo sheets and oil, reserving 1 teaspoon oil to brush on asparagus.
Spread ricotta mixture on phyllo stack, leaving a 1/2-inch border. Arrange asparagus spears side by side over ricotta mixture. Brush asparagus with remaining 1 teaspoon oil. Bake at 400°F until phyllo is browned and crisp, 22 to 25 minutes. Sprinkle with remaining 1/8 teaspoon salt; top with shaved cheese.