Add potato cubes to a medium saucepan and add enough water just to cover. Bring to boil; reduce heat slightly and leave at a low boil for 6 minutes. Transfer potatoes to a colander with a slotted spoon, reserving water. Rinse potatoes with cold water to slow their cooking.
Add more water to saucepan, enough so that it will cover the broccoli, and return to stove. Bring to boil. Add broccoli and blanch one minute. Drain and rinse with cold water. Drain thoroughly. Transfer to bowl of a food processor and pulse until finely chopped. Add chopped broccoli to medium bowl.
Transfer par-boiled potato cubes to food processor and pulse until finely chopped. Add to broccoli in bowl.
Add remaining ingredients to broccoli potato mixture and carefully stir with a fork to combine.
Using a mini ice cream scoop (like this one) or a teaspoon, scoop out 1 ½ tablespoon portions onto prepared baking sheet. You'll be able to fit about 25 on one baking sheet, so this recipe will make two batches. Shape each portion into a little log.
Bake 20 minutes, turning the baking sheet after 10 minutes.