I wouldn't recommend it. What's happening inside the meat while it's cooking, in a nutshell, is that proteins are denaturing, muscle fibers are contracting, water is being squeezed out of some areas into others, fats and tissue are rendering/gelatinizing--lots of activity in response to the heat. If you cut the thin piece off you stand a very good chance of allowing migrating juices to escape the thicker piece you leave behind.
Think of the thinner end as a protector for the thicker end. After the meat has rested (and the internal goings-on settle) you can remove the thinner piece, just treat it differently. Instead of slicing it chop it finely for use in hash, or mince it and heat it gently in unsalted butter as a topping for baked potatoes or as part of a filling for an omelette, or mix it into shredded braised cabbage.
If you'd rather, you can remove the thin piece before cooking and do something else with it but with something like brisket it cam be more trouble than it's worth.
