To prepare mayonnaise, combine first 4 ingredients, stirring until smooth; cover and chill.
To prepare ravioli, place shiitake mushrooms in a food processor; pulse 6 times or until finely chopped. Remove from processor. Place button mushrooms in food processor; pulse 10 times or until finely chopped.
Heat a large nonstick skillet coated with cooking spray over medium-high heat. Add mushrooms, shallots, thyme, and garlic; sauté 15 minutes or until tender. Add sherry; bring to a boil. Cook 8 minutes or until liquid almost evaporates. Remove from heat; stir in Parmesan, salt, and pepper. Place mushroom mixture in a bowl; let stand 10 minutes. Stir in sour cream.
Preheat oven to 425°.
Coat 2 baking sheets with cooking spray. Working with 1 wonton wrapper at a time (cover remaining wrappers to prevent drying), spoon 2 teaspoons mushroom mixture into center of wrapper. Moisten edges of wrapper with water; bring 2 opposite corners together. Press edges with a fork to seal, forming a triangle.
Place ravioli in a single layer on prepared baking sheets; spray tops lightly with cooking spray. Bake at 425° for 14 minutes or until golden brown, turning once. Serve with mayonnaise.