Preheat the oven to 375°F and lightly grease a loaf pan.
Heat the olive oil in a skillet over medium heat. Add the onion and garlic and cook for 3 minutes until softened.
Add the chopped mushrooms and 1 teaspoon of thyme to the skillet. Cook for 5 minutes until the mushrooms release their liquid and begin to brown. Remove from heat and let cool slightly.
In a large bowl, combine the ground beef, breadcrumbs, egg, salt, pepper, and the cooled mushroom mixture. Mix with your hands until just combined.
Press the mixture into the prepared loaf pan and sprinkle the remaining thyme on top. Bake for 45 minutes until the internal temperature reaches 160°F.
While the meatloaf bakes, rinse the wild rice and place it in a saucepan with the beef broth and butter. Bring to a boil, then reduce to low, cover, and simmer for 40 minutes until the rice is tender and the liquid is absorbed.
Let the meatloaf rest for 5 minutes before slicing. Serve slices alongside the wild rice pilaf.