Preheat your oven to 375°F (190°C) and lightly grease a 9x13 inch baking dish with olive oil or cooking spray.
Heat a large skillet over medium heat, add 1 lb of sliced sausage, and cook until browned on both sides, about 5-7 minutes. Remove the sausage and set aside.
In the same skillet, add the chopped potatoes and cook for 10 minutes, stirring occasionally. If needed, add a bit more olive oil to prevent sticking.
Add the diced onion and minced garlic to the skillet with the potatoes and cook for an additional 3-5 minutes until the onion is translucent and the garlic is fragrant.
Transfer the cooked potatoes, onion, garlic, and browned sausage to the prepared baking dish. Add 1 cup of asparagus pieces.
Pour 1 cup of chicken or vegetable broth over the mixture, season with 1 teaspoon dried thyme, salt, and pepper, and stir gently to combine.
Cover the dish with aluminum foil and bake in the preheated oven for 20 minutes.
Remove the foil, sprinkle 1 cup of shredded cheddar cheese over the top, and bake uncovered for an additional 15 minutes, until the cheese is bubbly and golden.
Let the casserole rest for a few minutes before serving. Garnish with fresh parsley if desired.