In a food processor, chop the almonds in the processor until they turn into small pieces. You're looking for some bits that are about the size of rice. It's ok if some of the almonds have turned powdery.
Next, toast the bread. Rip it into smaller pieces and chop them in the processor until they're about the size of peas or smaller.
Chop the broccoli florets in a food processor until they turn into small pieces. This shouldn't take too long.
In a large bowl, mix all the ingredients for the tots together.
Preheat oven to 350 degrees F (175 degrees C). You can prepare the tots two ways. Either line a large rimmed baking sheet with parchment paper, or spray a 24-hole mini muffin tin with cooking spray. I found that the muffin tin creates rounder tots, but the tots are more brown when cooked on the baking sheet.
Shape about 1 1/2 to 2 tablespoons of the broccoli mixture, and shape the into a ball. Repeat until all the mixture is rolled up. Set the tots into the muffin tin or on the baking sheet. Bake for about 20 to 23 minutes, or until the tops are browned.
While the tots are baking, prepare the cheese sauce. Heat a saucepan over medium-high heat, and add the olive oil. When the saucepan is hot, add the all purpose flour, and stir it with the oil immediately. The flour should start to bubble. Add the milk and let it boil. Reduce the heat to a medium low, and stir in the cheddar cheese. Remove from heat and mix in the onion powder, white pepper, and salt. The cheese sauce may harden as it cools, but just stick it in the microwave for 30 seconds, and it should be good to go.
Enjoy the tots with the cheese sauce!