Preheat the oven to 400°F (200°C).
Heat olive oil in a large skillet over medium-high heat.
Add ground beef and cook, breaking it apart with a spoon, until browned and cooked through, about 6-8 minutes.
Add diced onion, minced garlic, and diced carrots to the skillet. Cook until vegetables are softened, about 5 minutes.
Sprinkle whole wheat flour over the beef and vegetables. Stir well and cook for 2 minutes to remove the raw flour taste.
Gradually pour in beef broth while stirring constantly to avoid lumps. Bring to a simmer and cook until the mixture thickens, about 5 minutes.
Remove skillet from heat and stir in frozen peas, frozen corn, Greek yogurt, dried thyme, dried rosemary, black pepper, and salt. Mix until combined.
Transfer the beef and vegetable filling into a 9-inch pie dish and spread evenly.
Unroll the whole wheat pie crust and place it over the filling. Trim any excess dough and crimp the edges to seal.
Cut a few small slits in the top crust to allow steam to escape.
Brush the beaten egg over the crust to give it a golden finish.
Bake in the preheated oven for 35-40 minutes or until the crust is golden brown and the filling is bubbly.
Remove from the oven and let cool for 10 minutes before serving.