To prepare ravioli, place mushrooms in food processor; pulse 10 times or until finely chopped. Heat oil and butter in a large nonstick skillet over medium-high heat. Add shallots, and sauté for 2 minutes. Add mushrooms and 1/8 teaspoon salt; cook 5 minutes or until moisture evaporates, stirring occasionally.
Working with 1 wonton wrapper at a time (cover remaining wrappers with a damp towel to keep them from drying), spoon about 2 teaspoons mushroom mixture into center of each wrapper. Moisten edges of dough with water; bring 2 opposite corners together. Pinch edges together to seal, forming a triangle. Place ravioli on a large baking sheet sprinkled with cornstarch.
To prepare sauce, combine milk and flour in a small saucepan over medium-low heat; stir with a whisk. Cook 4 minutes or until slightly thickened, stirring frequently. Remove from heat; stir in cheese, 1 tablespoon chives, 1/8 teaspoon salt, and pepper. Set aside; keep warm.
Cook ravioli in boiling water 2 minutes or until tender. Drain. Serve with sauce. Garnish with fresh chives, if desired.