Preheat oven to 400°F. Line a large sheet pan with parchment.
Warm the olive oil in a skillet over medium heat. Soften onion and carrot, about 5 minutes.
Stir in garlic and cook until fragrant, 30 seconds.
Add lamb and brown, breaking it up. Spoon off extra fat if needed.
Mix in tomato paste and Worcestershire. Cook 1 minute to deepen flavor.
Whisk cornstarch into the broth. Pour into skillet with thyme, salt, and pepper. Simmer until thick, 3–4 minutes.
Fold in peas. Take off heat and let the filling cool 10 minutes.
Dust the counter with flour. Unfold puff pastry and gently roll to even thickness.
Cut each sheet into 4 rectangles for 8 pieces total. Stir egg with water for wash.
Spoon a heaping tablespoon of mashed potatoes onto each piece. Top with a few spoonfuls of lamb filling.
Brush edges with egg wash. Fold to enclose, crimp with a fork, and cut small vents on top.
Arrange on the pan, brush tops with more egg wash, and bake 22–26 minutes until deeply golden.
Cool 5 minutes before serving to set the filling.