1. Preheat the oven to 400 degrees F. Melt the butter in a large pot over medium heat. Add the onion and saute 3 minutes. Sprinkle onion with the flour and cook 2 minutes. Add the celery, carrots, peas and chicken broth. Turn heat to medium high; cook 2 to 3 minutes, until sauce thickens. Add the chicken, salt and basil to taste; mix.
2. Spoon the mixture into six oven-safe ramekins, filling each to the top.
3. Unfold the sheets of phyllo dough, keeping them stacked. Use a paring knife to cut the dough into six circles that are 1 1/2 inches bigger in diameter than the ramekins. Place one stack on each ramekin; brush with some of the egg yolk. Top with the 1/4 teaspoon of basil and coat again with egg yolk.
4. Cook on the top rack of oven 10 minutes. Serve.