Preheat oven to 375°F (190°C).
Melt the butter in a deep skillet over medium heat. Add the mushrooms, onion, tarragon, salt and pepper. Cook for 5 minutes, stirring often or until softened. Stir in the lemon juice.
Whisk the milk with the corn starch. Gradually stir the corn starch mixture into the mushroom mixture. Bring to a boil, stirring until thickened. Stir in the chicken and asparagus tips and remove from the heat.
Butter six 1-cup (250 mL) pans (or ramekins). Transfer the filling mixture to prepared pans. Unroll the crescent dough and seal the perforations using fingertips. Cut into six squares. Gently stretch each square of dough to cover each pan, allowing the edges to overhang. Brush with beaten egg.
Bake for about 15 minutes or until top is golden and filling is warmed through.