Adjust oven rack to lower-middle position and heat oven to 300 degrees. Pat beef dry with paper towels and sprinkle with salt and pepper. Heat 2 tablespoons oil in Dutch oven over medium-high heat until just smoking. Brown beef on all sides, 5 to 10 minutes; transfer to bowl.

Add mushrooms, onion, and remaining 2 tablespoons oil to fat left in pot, cover, and cook over medium heat until mushrooms have released their liquid, about 5 minutes. Uncover and continue to cook until mushrooms are dry and onion is lightly browned, 5 to 10 minutes. Stir in tomato paste and thyme and cook until fragrant, about 1 minute.

Stir in broth and bay leaves, scraping up any browned bits. Stir in beef and any accumulated juices and bring to simmer. Cover, transfer pot to oven, and cook for 2 hours. Stir in cauliflower and continue to cook until meat and vegetables are tender, about 30 minutes.

Remove stew from oven and discard bay leaves. Transfer 1 cup cooking liquid and 1 cup cauliflower to blender and process until smooth, about 30 seconds, scraping down sides of blender jar as needed. Stir cauliflower puree and vinegar into stew and season with salt and pepper to taste. Sprinkle individual portions evenly with parsley before serving.