Put the carrot, onion, mushrooms, garlic, and parsley in a food processor fitted with a metal blade. Pulse to chop the vegetables into a tiny dice that looks a little like confetti (about 30 pulses). Stop and scrape down the sides as needed.
Heat the olive oil in a large skillet over medium-high heat. Add the chopped vegetables, thyme, paprika, half the salt, and pepper and cook, stirring frequently, until the juices of the vegetables release and cook off, about 8 minutes. The mixture should look slightly moist but not wet and the pan should be dry.
Make a panade with the oat milk and breadcrumbs. Let it soak for about 5 minutes before mixing it into the meat substitute.
Transfer the cooked vegetables to a large bowl and let cool slightly. Add the meat, salt, panko breadcrumbs, tomato paste, rest of the spices and egg. Use your hands to mix the ingredients thoroughly until everything is evenly distributed.
Transfer the mixture to a 1-pound (9x5-inch) loaf pan and press it gently to fill out the pan. Bake until the meatloaf is firm to the touch, about 50 minutes.
In a small bowl, stir together the ketchup, mustard and brown sugar. Use the back of a spoon to spread it evenly over the top of the meatloaf. Return the meatloaf to the oven and bake to finish cooking the meatloaf and darken the glaze, another 15 minutes.