Thinking about cooking beef tongue is kind of a wayback machine. I've eaten about a metric butt-load of beef tongue, but I've never cooked it. So what I say about cooking tongue is rumor and innuendo.
My Mom cooked it about once a month in the '60s and '70s. We had Basque friends, and she learned how to cook it from them. Mom would simmer it for hours, then peel it. Then she would take it back to the butcher and he would cut the tongue into very thin slices. So it was cooked at that point, and Mom could do whatever she wanted with it. Most of the tongue would get quick-pickled and served as a side dish. The rest of the tongue was fried in Crisco in a cast-iron skillet for sandwiches.
And the only way I've ever had tongue was sliced very thinly. Basque restaurants here serve it pickled as a side. Lengua that I have seen is also thinly sliced.
But I would think that you could braise or boil it, peel it, then finish it on the grill with great results. But you might think about thin slices.