Scrub potatoes and cut into quarters. Peel carrots, and trim and discard ends; cut carrots diagonally into 1/4-inch-thick slices. Rinse beans, and trim and discard ends; cut beans into 1 1/2-inch lengths.
In a covered 5- to 6-quart pan over high heat, bring potatoes, carrots, broth, and herbs to a boil. Reduce heat and simmer until carrots are almost tender when pierced, about 5 minutes. Add beans; cover and simmer until all the vegetables are tender when pierced, 3 to 5 minutes longer.
In a small bowl, mix cornstarch and cream. Add to vegetable mixture and stir until boiling, about 2 minutes. Remove from heat and stir in chicken. Add salt and pepper to taste. Spoon equally into 6 round soufflé dishes or ramekins (1-cup size; 3 1/2 to 4 1/2 in. wide). Let cool to room temperature, about 35 minutes.
Meanwhile, on a lightly floured board, roll each puff pastry shell into a round about 1 inch wider than the diameter of soufflé dishes. Brush egg around edge of each pastry in a border about 1/2 inch wide. Invert a pastry onto each dish, egg side down, and press edges firmly against sides of dish. Brush egg lightly over pastry tops and sides. Set dishes at least 1 inch apart in a 10- by 15-inch pan.
Bake in a 400° regular or convection oven until pastry is richly browned, 15 to 25 minutes.