Rinse the red lentils under cold water and drain them.
In a medium saucepan, combine the red lentils and water. Bring to a boil over high heat, then reduce the heat to low and simmer, covered, for about 15-20 minutes, or until the lentils are tender and the liquid is absorbed. Remove from heat and let cool slightly.
In a small bowl, mix together the ground flaxseed and 3 tablespoons of water. Let it sit for about 5 minutes to thicken and form a flax "egg." Set aside.
In a skillet, sauté chopped onion for 2 -3 minutes in a tablespoon of oil. Add carrot and garlic and sauté for 2-3 more minutes. Set aside.
In a large mixing bowl, mash the cooked lentils with a fork or potato masher until mostly smooth but still slightly chunky.
Add the sautéed chopped onion, minced garlic, grated carrot, rolled oats, flax "egg," nutritional yeast (if using), ground cumin, smoked paprika, pinch of cayenne, salt, and pepper to the mashed lentils. Stir until well combined.
Shape the lentil mixture into patties of your desired size and thickness.
Heat a drizzle of olive oil in a large skillet over medium heat. Once hot, add the lentil patties to the skillet (you may need to do this in batches depending on the size of your skillet).
Cook the patties for about 4-5 minutes on each side, or until golden brown and crispy.
Once cooked, remove the lentil patties from the skillet and place them on a paper towel-lined plate to drain any excess oil.
Serve the Vegan Red Lentil Patties hot with your favorite toppings, such as avocado slices, lettuce, tomato, and vegan mayo, or enjoy them plain as a tasty protein-packed snack.