Start by chopping the onions. You want to chop them so they are a bit bigger than you would usually chop them for a soup. The onions will break down a bit as they caramelize, so you want chunks of onions so that they don't completely fall apart once you add them to the soup.

Melt a tablespoon of butter in the bottom of a large pot or dutch oven. Once the butter melts, add the chopped onion and stir to evenly coat in the butter. When I caramelize onions I like to speed up the process a little by adding ½ cup of water, turning the heat to medium high, and covering the pot. This softens the onions faster. Then once the onions are soft and the most of the water has evaporated, I turn the heat down to medium low. Keep and eye on the onions and stir frequently. After about 35-45 minutes the onions will be golden brown. Pour them into a bowl and set aside.

Next you need to brown the beef. Pour the flour into a bowl and season it with salt and pepper. Melt a tablespoon of butter in the same pot that you used to caramelize the onions, over medium heat. Once the butter has melted, coat the beef in the flour and add half of it to the pot. Let it cook for about 5 minutes, flipping it a couple times, until it has browned. Once it has browned remove it from the pot and set aside. Add another tablespoon of butter to the pot and brown the rest of the beef, and then set it aside.

Using the same pot, melt the remaining butter over medium heat. Once the butter has melted, add the mushrooms and garlic. Stir frequently and cook the mushrooms until they have lightly browned. Then add the carrots, celery, and thyme. Sauté for another 4 minutes.