In a skillet heated to medium, add 1 tablespoon of the oil and sauté the carrots for 20 minutes, until easily pierced with a fork but firm, not mushy.
Add the mushrooms and onion and sauté over medium heat for 5 to 10 minutes, until softened and browning a bit. Transfer to a food processor with the remaining ingredients. Pulse together 30 to 35 times, until just broken up and sticky with texture and bits of color intact. Taste—if needed, season with more salt or seasonings. Pulse or stir to incorporate.
Form 1½-inch (4 cm) meatballs with your hands. Heat a skillet to medium and add the remaining oil. Slow-cook the meatballs, rotating often, for 10 to 15 minutes, until browned on all sides. Serve warm.