Add carrots, celery, onion, and thyme. Cook, stirring occasionally, until vegetables are tender, about 10 minutes.
Add flour and tomato paste and stir for 1 minute.
Add beef and cook until browned. Add 1/2 cup water, bring to a boil, and simmer for 1-2 minutes. Season to taste with salt and pepper.
Separate the biscuits into rounds and roll them out flat. Put a generous spoonful of meat in the center of each one and fold them in half, pressing the edges together with a fork. Place on a cookie sheet lined with parchment or a silicone mat and brush with egg.
Bake until golden, about 15 minutes. Cool and serve.