At least a day before you plan to make the tots, cook the potatoes (whole with skin left on) in the Instant Pot until just tender (about 15 minutes), then cool in the refrigerator at least 24 hours. Steam the broccoli until soft enough to fall apart easily. This can be done in the Instant Pot in a steamer basket with 3/4 cup water in the bottom and cooked for 1 minute with QUICK release. Chill the broccoli until you're ready to make the tots.
Shred the potatoes with a grater into a large bowl. You can save the butt ends of the potatoes to bake into potato skins. Don't include them in with the shredded potatoes. Chop the cooked broccoli with a fork and gently fold it into the shredded potatoes along with the seasonings.
Preheat oven to 400 degrees, then line a large baking tray with parchment paper. Take 2 T of the mixture and roll it into a log shape, flattening the ends. Place on tray. Do this until the mixture is used up. Bake for about 45-50 minutes, flipping them at the halfway mark.
Serve with ketchup, Sriracha, or any other sauce that sounds yummy.