Place the beef liver on a cutting board and trim away any tough membranes, connective tissue, or large veins. Cut the liver into evenly sized pieces approximately 1 to 2 inches wide.
Pat the liver dry with paper towels. Season the pieces evenly with the salt.
Heat the beef tallow in a heavy skillet over medium heat. Allow the fat to melt and become hot, but do not let it smoke.
Add the liver pieces in a single layer. Cook for approximately 3 to 4 minutes on the first side, then turn and cook for another 2 to 4 minutes.
Cook the liver until it is browned on the outside and just cooked through inside. Avoid cooking it until it becomes dry or hard, as overcooked liver can make the finished pate grainy.
Remove the skillet from the heat and allow the liver to rest for approximately 5 minutes.
Transfer the warm liver and any cooking juices to a food processor. Add 6 tablespoons of the softened butter and the optional thyme.
Blend until the mixture begins to become smooth. Stop and scrape down the sides of the food processor with a rubber spatula.
Continue blending while adding warm water, melted butter, or cooking juices 1 tablespoon at a time. Process until the pate is creamy, smooth, and easily spreadable.
Taste the mixture and add more salt when needed. Blend once more to distribute the seasoning evenly.
Spoon the pate into small ramekins or an airtight container. Smooth the surface with the back of a spoon.
Melt the remaining 2 tablespoons of butter and pour a thin layer over the surface of the pate. This helps protect the pate from air and prevents the top from drying out.
Serve the pate warm and soft, or cover and refrigerate it for at least 2 hours for a firmer texture.
Remove chilled pate from the refrigerator approximately 15 to 20 minutes before serving to make it easier to spread.