The best tongue tacos I've ever eaten came out of a Mexican restaurant run by a husband and wife, not far from where I once lived in California. It was a small joint, bustling during the lunch hour and almost empty in the evenings. I'd show up at the end of the day in need of a steadying meal and comfort in any guise. Sometimes it was very quiet in there. On summer days there was nothing but the whirr of fans overhead and the sound of the cook's metal spatula brushing up against the griddle.
I never had a single bad meal there, nor did I ever have a meal that could rightly be called innovative or exceptional. But every dinner I ate there was delicious in a quiet way. The soups simmered with tomatoes and onions and fresh herbs never seemed watery or dull. The fish tacos were fried lightly so that the flesh was tender and salted just so, and so on. Dishes were always just right, which is what you would call know-how and patience in practice.
I remember the tacos for being just right. The father also served tongue simmered in tomatillo sauce, a mouth-puckering and tart gravy of sorts for the organ. The tongue for that preparation was very tender and soft. For tacos, the father took the same simmered tongue and charred it over the griddle. He did so with care and patience so that the char produced was not a surface-level browning, but a truly crisp exterior.
The browning is key to my enjoyment of tongue tacos, much like the charring of a burger patty. Without it, tongue is an awfully rich and soft experience, and biting down into a taco filled with it is no fun.
The mushiness of tongue is not uncommon in most people's tongue tacos experiences. Too many tacquerias just want to move your order along, so the cooks retrieve a chunk of tongue from their stash of animal parts, cut it up on the griddle, and cook it through until it is just warmed through and beginning to have a bit of color. Then they sweep the lengua into the awaiting tortillas, and it's almost like the griddle time never happened. This is just a sad state of affairs (and can also apply to ears, snout, lips, stomach, and everything else that is rich and in need of crisping up). I want a little bit of crispiness in my bite of lengua, along with the tartness of the lime, the freshness of the salsa and cilantro.
You could try to ask the nice man (or woman) behind the griddle to properly crisp up your tongue for you. Or, you can do it at home.
If you have a simmered tongue in your fridge—and getting to that stage takes no more effort than simmering a tongue in a whole pot with onions and some herbs—then you're close to having dinner on the table. At home, you have the freedom to use pork or lamb's tongue in lieu of the more classic beef or veal tongue. You cut up the tongue, put down a pat of oil or lard in your pan, and stand over the skillet until the tongue is ready to be moved onto the tortilla, and not one second before. Simple as that.