The molleja is one of the main components of the Sunday asado in Argentina. There are two types of sweetbreads: those of the throat and those of the heart (thyme); those of the throat are very lean while the thyme is much fatter. Needless to say, the thyme is much better!
Make sure that the sweetbreads are very fresh, because they usually have a very low shelf life. Start by preparing the grill for direct cooking, but with a low temperature, because the sweetbreads must cook slowly. Clean the sweetbreads from all the external fat, leaving only the one inside.
Sprinkle them with oil and place them on the grill. In the meantime, cut the lemons in half and put them on the grill pulp-side down.
Let the sweetbreads cook for about 20 minutes, turning them a couple of times. Once browned well outside, cut them into slices about half a centimeter thick and then put the slices back on the grill.
Give a light brush of oil on both sides, add a little salt and a nice sprinkling of juice from the grilled lemons.
Keep them grilling as you like while you make a tequila turn with salt and lemon remaining.
Enjoy your molleja!