Allow steaks to rest at room temperature for 15 - 20 minutes to take chill off.
Preheat a 3-qt. sear sauté pan over medium heat for about 1 minute. Season steak liberally with salt and pepper. Add olive oil and heat in pan.
Place steaks in pan and sear for 3 - 5 minutes per side or to desired doneness. Remove from pan and set on a plate to rest.
Add mushrooms to the sauté pan and caramelize for 2 - 3 minutes. Add a cube (or two) of butter if pan looks dry.
Add shallot and sweat for 1 minute.
Add garlic and cook for 1 more minute.
Add red wine and deglaze (using a wooden or nylon utensil to scrape bottom of pan to release fond or tasty bits that may have adhered to pan) and reduce by half.
Add beef stock and chopped thyme and continue to reduce by half on a simmer.
Whisk in cold butter and continue to simmer. Once butter has been incorporated, do not re-boil sauce. Keep warm on low.
For eggs, preheat a 10-in. omelette pan. Crack each egg into a small bowl and pour 2 eggs into the pan at a time, season with salt, pepper and crumbled blue cheese; cook for 1 minute and flip, using a nylon spatula.
Cook for 30 seconds and remove from pan and keep warm.
To plate, place a steak on each plate and top with mushroom sauce and one fried egg.