Pulse the rice, lentils, and beets together in a food processor. Pulse until the veggie mixture begins to come together. The mixture should stick to itself and begin to form a ball, but it still has some chunky texture. Depending on your food processor, this can take 10 to 25 pulses. Place the mixture in a large mixing bowl.
Grind the porcini mushroom into a fine powder in a spice or clean coffee grinder. Removing any large pieces, add the mushroom powder to the beet mixture. Add all of the remaining ingredients, and with your hands, incorporate everything until it is thoroughly combined.
Form the mixture into 6 round patties, or for larger burgers, 4 patties. If the burgers are going to be grilled, the larger patties are recommended. Allow the patties to chill in the refrigerator for at least 30 minutes or overnight if made in advance.
To cook indoors, heat a large, heavy-bottomed nonstick or cast-iron skillet to medium-high heat, and spray or drizzle the pan with a thin layer of oil. Cook the beet burgers for 5 to 6 minutes on each side, or until they are dark brown and crispy around the edges.
To grill, brush burgers with a thin layer of oil and grill, checking after 3 minutes, until brown and charred on both sides, about 6 minutes. Serve with your favourite burger fixings and sides.