Follow the steps in the Dependent Recipes section to make the mashed potatoes. For this recipe, there really can never be enough mashed potatoes, so feel free to make 1.5X the mashed potatoes (you will use them :-).
MAKING THE GROUND BEEF AND VEGETABLES
Cook the ground beef until it's cooked/browned, drain any fat, set aside.
Add olive oil to a large saucepan. Saute the carrots until they begin to get tender.
Add in the onions and saute until they begin to become translucent.
Add in the cooked ground beef.
Season with black pepper and thyme.
Add the butter and peas.
Sprinkle the flour over the pan and then stir this in.
Add tomato paste, wine and the Worcestershire sauce. Let this reduce, cooking off the alcohol from the wine.
Add the chicken stock. Allow this to reduce down until it begins to get quite thick.
ASSEMBLING THE PIE
Grease an oven-proof baking pan using some butter. I used a 9x9 pan and it was perfect, though a 9x13 might work for a thinner pie.
Spoon the mashed potatoes on top, flatten them out with a spoon.
Glaze the top with the egg.
Bake for approximately 20 minutes or until the top of the potatoes begins to brown.