Preheat oven to 400°. Place cauliflower on a large sheet pan. Sprinkle with ½ tsp salt and ¼ tsp pepper. Toss with 2 tbsp olive oil. Spread out on sheet pan and top with Parmesan cheese. Bake for 30 minutes or until tender and browned.
Meanwhile, roughly chop onion and celery, and mince garlic. Chop mushrooms into small pieces.
Heat 2 tbsp olive oil in a large skillet over medium heat. Add onion, celery, and garlic and cook briefly until fragrant. Add mushrooms and cook, stirring occasionally, until starting to soften, about 2-3 minutes.
Add ground beef, ½ tsp salt, and ¼ tsp pepper to the skillet and cook until beef is no longer pink and mushrooms are starting to brown, about 5-6 minutes.
Add tomato sauce, Worcestershire, and chipotle peppers. Reduce heat as needed to maintain a simmer and cook, stirring occasionally, for about 4-6 minutes or until sauce is reduced and thickened (you can cook more or less depending on how sloppy you like your joes).
Spread butter or mayo on buns and toast in the oven for a few minutes until warm and lightly toasted.
Fill buns with sloppy joe filling. Crumble goat cheese on top of filling. Serve with roasted cauliflower.