In a large bowl, combine hash browns, carrot, pepper, mushroom, onion, garlic, and corn. Season with salt and pepper. Add egg and flour and mix together to form batter.
Heat 1 tablespoon of vegetable oil in a large skillet over medium heat. Drop batter, about 1/4 cup at a time, onto skillet and gently flatten into circles (you will be able to cook 3-4 at a time). Cook for about 4 minutes on each side, until crisp and golden brown.
Repeat with the remaining batter and oil.
Meanwhile, in a small bowl, whisk together dipping ingredients. Serve with warm pancakes.