Bring the water and salt to a boil in a small saucepan. Remove from the heat, stir in the bulgur, and cover. Let it stand until the grains are tender, about 15 to 20 minutes. Drain any excess water and spread the bulgur onto a rimmed baking sheet to cool slightly.

Pulse the shredded beet, walnuts, basil, and garlic in a food processor until finely chopped, about 12 pulses, scraping down the sides of the bowl as needed.

Add the beans, carrot baby food, water, mustard, salt, and pepper to the food processor. Pulse until the mixture is well combined but still retains some texture, about 8 pulses.

Transfer the mixture to a large bowl and stir in the panko and the cooled bulgur. Divide the mixture into 8 equal portions and pack them firmly into 4-inch-wide patties.

Heat half of the oil in a 12-inch nonstick skillet over medium-high heat until the oil is shimmering. Gently lay 4 patties in the skillet and cook until crisp and well browned on both sides, 4 to 5 minutes per side. Use a wide spatula to turn them gently halfway through. If the burgers begin to scorch, reduce the heat to medium.

Transfer the cooked burgers to a plate and tent loosely with foil to keep warm. Wipe out the skillet with paper towels and repeat the process with the remaining oil and patties. Serve hot.