Preheat your oven to 400°F (200°C).
Place the cubed potato and cauliflower florets in a pot of boiling water and cook for 15 minutes until very tender.
While the vegetables boil, heat the olive oil in a large oven-safe skillet over medium-high heat.
Add the diced onion, carrots, and chopped mushrooms to the skillet and sauté for 5-7 minutes until the vegetables have softened.
Add the ground beef to the skillet, breaking it up with a wooden spoon, and cook until fully browned.
Stir in the tomato paste, sea salt, black pepper, and dried thyme, cooking for 1 minute to allow the flavors to meld.
Pour in the beef broth and add the frozen peas, then simmer for 5 minutes until the liquid has slightly reduced.
Drain the cooked potato and cauliflower, then mash them together until the texture is smooth and velvety.
Spread the mash evenly over the beef and vegetable mixture in the skillet, smoothing it with a spatula.
Transfer the skillet to the oven and bake for 15-20 minutes until the topping is lightly browned and the filling is bubbling.