Preheat oven to 375°F. Lightly spray a 9x13-inch baking dish with nonstick cooking spray.
In a large skillet over medium heat, melt the butter. Add onions, carrots, potato, celery, salt, and pepper. Cook until the vegetables are soft (about 5-8 minutes) stirring occasionally.
Sprinkle the flour over the vegetables and stir to combine. Cook for 1 minute.
Pour in the chicken stock and milk. Stir to combine. Bring the mixture to a boil, and then reduce heat to a simmer. Add chicken and continue to simmer for 3-4 minutes, or until it thickens slightly.
Add frozen peas and corn. Stir to combine.
Pour the mixture into the prepared baking dish. Top with 8 dollops (about ½ cup each) of biscuit batter, and flatten slightly.
Brush tops of biscuits with the beaten egg.
Bake, uncovered, for 30-35 minutes, or until the biscuits are golden brown and the casserole is hot and bubbly.
Garnish with parsley and serve.