1. Preheat oven to 350°F; line a large baking sheet with parchment. Place phyllo shells in a single layer on lined baking sheet.
2. Combine butter, both sugars and heavy cream in a large saucepan. Stir over high heat until blended and beginning to boil. Reduce heat to medium and boil gently, stirring often, for 2 minutes. Remove from heat and carefully stir in nuts and vanilla. Work carefully; mixture is extremely hot and will bubble up. Let cool slightly.
3. Divide filling among tart shells, using about 1 heaping tsp. in each (do not overfill). Bake until shells have crisped and are light golden, 10 to 15 minutes. Let cool before serving.