Preheat the oven to 350°F. Grease the wells of a 12-cup muffin tin with olive oil.
Heat one tablespoon olive oil in a medium skillet over medium heat and sauté the onions, mushrooms, and bell peppers for about two minutes. Remove from the heat and allow the vegetables to cool.
In a small bowl, combine the tomato paste, mustard, worcestershire sauce, rosemary, garlic, and pepper.
In a large bowl, mix together the beef, the reserved sautéed vegetables, egg, thyme, and garlic powder. Stir in half of the tomato mixture and combine well.
Spoon the meat mixture into the greased muffin tin and bake for 10 minutes.
While the muffins are baking, combine the remaining tomato mixture, carrot juice, and water in a small saucepan. Bring to a quick boil over medium-high heat, and then reduce the heat and simmer, covered, until the muffins are removed from the oven.
Remove the muffins from the oven and spoon or brush the sauce on each muffin. Place the muffins back in the oven and bake for another 15 minutes, or until the internal temperature is 160°F.
Allow the muffins to stand for about five minutes before eating.