Place the beef cubes, carrots, mushrooms, onion, and garlic in the slow cooker.
Add the red wine, beef broth, tomato paste, thyme sprigs, bay leaves, 1 teaspoon salt, and black pepper. Stir to combine.
Cover and cook on low for 8 hours until the beef is very tender.
About 30 minutes before serving, place the potatoes in a large pot and cover with water. Bring to a boil and cook until fork-tender, about 15 minutes.
Drain the potatoes and return them to the pot. Add the butter, milk, and 1/2 teaspoon salt. Mash until smooth.
Remove the thyme sprigs and bay leaves from the slow cooker. Serve the beef bourguignon over the mashed potatoes.