Preheat the oven to 225 degrees.
Pat the beef dry and season well with salt and pepper. Place the beef cubes and flour in a gallon-sized resealable bag. Shake everything well to coat.
Place a large oven-proof skillet with a lid over medium-high heat and heat 2 Tbsp olive oil. Add as many beef cubes as will fit comfortably into the skillet, but be sure not to overcrowd the pan. Sauté for about 1–2 minutes per side until golden brown. Remove the beef from the skillet to a plate and cover lightly with foil. Repeat with remaining beef.
When all of the beef is cooked and set aside, add the onions, mushrooms and ½ tsp salt to the skillet. Cook for about 5 minutes, stirring occasionally, until the vegetables are softened. Add garlic and cook for 1 additional minute.
Pour ½ cup chicken broth into the skillet and stir to scrape up anything that is sticking to the bottom of the pan. Once the broth has nearly all evaporated, add ¾ cup more chicken broth and the thyme. Bring the mixture to a boil and add the beef back to the pan. Cover and transfer to the preheated oven. Let the beef cook slowly for about 2½ hours. The beef will be ready when it is fall-apart tender so leave it in the oven until it is ready.
Remove the pan from the oven and strain the liquid from the pan into a measuring cup. Add ½ cup of the liquid back to the pan along with the beef and stir in the milk. Sprinkle the Asiago cheese over the top of the sauce and stir constantly until the cheese melts and the sauce has slightly thickened. Sprinkle with a bit of Parmesan cheese and serve right away, either all by itself, or spooned over pasta or mashed potatoes. Enjoy!