Chop onion, carrot, celery, and garlic finely. Melt butter in a saucepan over medium heat, then add the chopped vegetables. Saute in butter until soft and lightly browned, approximately 5 minutes.
Sprinkle flour over the vegetables and stir well until it forms a roux (thick paste-like consistency). Continue cooking the roux until it turns a golden-brown color - don't let it get too dark!
Slowly pour in red wine while stirring continuously. Scrape up the browned bits from the bottom of your pan with a wooden spoon, and let the wine simmer for a few minutes.
Add beef stock, tomato paste, bay leaf, and thyme, and stir to combine.
Bring to a simmer, then reduce to low. Let it simmer gently for about 1 hour, stirring now and then.
After cooking, remove the bay leaves and thyme, then pour the sauce through a fine-mesh strainer over a mixing bowl. Press down on the vegetables to extract as much of the liquid as possible.
Rinse the saucepan, then return the espagnole sauce to the pan. Season with salt and pepper to taste, and simmer an additional 5 minutes before serving! Enjoy!