In the Instant Pot, combine the chicken, onion, garlic, and cilantro. Pour the salsa verde and chicken broth on top; season with the salt.
Lock the lid into place and set the steam release valve to sealed. Select Manual or PRESSURE COOK and set the timer for 15 minutes on high.
When cooking is complete, allow the pressure to release naturally. Unlock and remove the lid.
Discard the cilantro sprigs. Using two forks, shred the chicken in the pot.
Heat the corn tortillas on a comal or griddle over medium-high heat until soft and pliable. Spoon 2 to 3 heaping tablespoons of the shredded chicken mixture down the center of each tortilla. Roll the taquitos up tightly, securing with a toothpick if necessary.
Heat the vegetable oil in a medium skillet over high heat. Fry the taquitos, 3 to 4 at a time, until golden and crisp. As you finish frying each batch, transfer the taquitos to a paper towel–lined plate to drain any excess oil.
To serve, arrange 2 or 3 taquitos per plate. Garnish with shredded lettuce, guacamole, and queso cotija.