Cut meat into 1-inch strips, and place in a medium bowl. Toss with salt, cover loosely with plastic wrap, and refrigerate for 18 to 24 hours.
Assemble a meat grinder with a 3/16-inch die and a sharp knife. Refrigerate grinder until thoroughly chilled (this will ensure proper texture and flavor).
Pass meat through the grinder twice. On the first pass, pay special attention that the meat is emerging cleanly and evenly. If it isn't, turn the grinder off immediately, and remove and clean the blade and knife. Reinstall and finish grinding the meat. As an alternative to grinding the meat, you can finely chop it by hand or in a food processor with a very sharp blade, however, this will result in a more fragile burger.
Working quickly, divide beef into four portions, and form into 3/4-inch-thick patties. Press patties a bit thinner in the center; the meat shrinks as it cooks and the patties will emerge an even thickness only if they start out thinner in the center. The patties may be made ahead up to this point and refrigerated, loosely covered, up to 12 hours ahead.
To cook the hamburgers on the stovetop: Choose a skillet (preferably cast-iron) not much larger than will hold the patties. Heat the pan over medium heat until the meat sizzles on contact. Cook until golden brown, about 3 minutes per side. Reduce heat to medium-low, and continue to cook, flipping patties twice, 4 minutes more for medium-rare. Remove from heat, and let rest for 2 to 3 minutes before serving.
Serve on foccacia with aioli and lettuce; serve pickles on the side.
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