Load the Instant Pot. Place the chicken in the Instant Pot along with the green chilis, onion, garlic, ranch seasoning, and chicken broth. Give everything a good stir. Scatter the butter over it all.
Cook. Seal the Instant Pot and cook on high pressure for 40 minutes. Manually release the pressure.
Shred. In the Instant Pot, shred the chicken with 2 forks.
Finish it off. Stir in the ranch dressing.
Toast the buns. Halve the buns and arrange them, cut side up on a baking sheet lined with aluminum foil. Brush the buns with butter and pop them under the broiler for a minute or two until lightly golden brown. Remove the tops of the buns and set them aside, leaving the bottoms on the baking sheet.
Assemble the sandwiches. Divide the chicken between the toasted bun bottoms and sprinkle the cheese over the tops.
Melt. Place the open sandwiches under the broiler for a minute or so until the cheese has melted.
Close. Top the sandwiches with the top buns.
Notes
To easily shred the chicken, leave it in the Instant Pot or transfer it to another bowl and grab your hand mixer or electric mixer, and on a low-speed turn on the mixer and it will instantly begin shredding the chicken to your desired texture.