1. To prepare filling, heat a skillet over medium-high heat. Coat pan with cooking spray. Add shallots and mushrooms to pan; sauté 6 minutes, stirring frequently. Add wine to pan; cook 5 minutes or until liquid almost evaporates. Remove from heat. Stir in cheese, thyme, and 1/2 teaspoon salt. Cool to room temperature.
2. Preheat oven to 350°.
3. To prepare pastry, place walnuts in food processor; process until smooth, scraping sides of bowl. Weigh or lightly spoon flour into dry measuring cups; level with a knife. Add flour, 1/2 teaspoon salt, and butter to food processor; pulse to combine. Place flour mixture in a medium bowl. Add 1/4 cup ice water and 1 egg white to bowl; stir until moist. Turn dough onto a lightly floured surface. Divide dough into 18 equal portions; roll each portion into a ball. Working with one ball at a time (cover remaining dough to prevent drying), roll ball to a 3-inch circle. Spoon about 2 teaspoons mushroom mixture in center of circle. Fold dough over filling; crimp edges with a fork to seal. Place on a baking sheet lined with parchment paper. Repeat procedure with remaining dough balls and filling, placing turnovers about 1 inch apart on baking sheet. Combine remaining egg white and milk in a bowl. Lightly brush turnovers with milk mixture.
4. Bake at 350° for 16 minutes. Cool on wire rack.