Heat the oil and add the onions and garlic along with some salt and ground black pepper. Saute for a couple of minutes until the onion start to brown.
Add the thyme and vegan sausage and continue to saute a couple more minutes. Add the hash browns and stir to mix.
Turn off the heat and stir in the cream of mushroom soup and the mozzarella shreds. Reserve some of the mozzarella shreds to sprinkle over the top, if you wish. Before you sprinkle the mozzarella over the top, make sure you check for seasoning and add more salt and pepper to your taste. (If you aren't using a cast-iron skillet, coat a 9 X 12 baking dish with cooking spray and turn the mixture into the dish, then sprinkle the remaining mozzarella over the top.)
Place the skillet or the baking dish into the oven and bake 45 minutes until the cheese is all bubbly and golden.