In a large pot, combine the pork, jalapeños, 1 teaspoon salt and 7 cups water. Bring to a boil over high, then reduce to low, cover and cook until a skewer inserted into the pork meets no resistance, 60 to 75 minutes.

Using a slotted spoon, transfer the pork to a medium bowl; set aside to cool slightly. Meanwhile, pour the broth through a fine mesh strainer set over a large bowl and let cool.

When the pork is cool enough to handle, shred into bite-size pieces, discarding any fat and gristle. Cover both the broth and shredded meat and refrigerate until ready to use or up to 3 days.