Combine the ground beef, pureed liver, egg, and salt in a medium bowl. Mix thoroughly with your hands until fully combined. Divide the mixture into 4 equal portions and shape into patties about 4 inches in diameter.

Heat a large cast-iron skillet over medium-high heat for about 5 minutes, until it begins to smoke lightly. Add the tallow, swirling to coat the pan evenly.

Place the burger patties in the skillet and cook for 3 to 4 minutes per side, or until a crust forms and the burgers are medium done (internal temperature of about 140°F). For well-done burgers, cook for about 5 minutes per side, or until the internal temperature reaches 160°F.

Remove from heat and serve immediately with your favorite burger toppings and buns, or enjoy as-is for a low-carb option.