Preheat the oven to 375°F (190°C).
In a large pan, melt butter over medium heat and sauté onions and garlic until fragrant.
Add flour and stir continuously for about 1 minute to form a roux.
Gradually whisk in vegetable broth and milk, stirring to avoid lumps.
Add potatoes, carrots, and mixed vegetables. Cook until vegetables are slightly tender.
Stir in sweet corn, salt, pepper, and thyme. Let simmer for 5 minutes.
Pour the mixture into a greased baking dish and cover with the pie crust.
Brush the top with beaten egg for a golden finish.
Bake for 25-30 minutes or until the crust is golden brown.
Let it cool slightly before serving.