Cut cauliflower into medium-sized florets and process in batches in a food processor.
For each batch, pulse until pieces break down and continue processing until texture resembles rice grains. Transfer to a large pot.
Sprinkle with kosher salt.
Cook over medium-high heat, stirring, until mixture is wet and soft, about 30 minutes. Reduce heat if it begins to brown. Remove from heat and allow cauliflower to cool.
Transfer cauliflower in batches to a clean cotton towel. Bring up sides, twist towel, and squeeze out as much liquid as you can. For this recipe, you should have about 3 1/2 cups cooked, squeezed-dry cauliflower.
Preheat oven to 350 degrees F.
Line a large rectangular baking dish with parchment paper, allowing ends to extend up over the sides for easy lifting.
Lightly brush parchment with olive oil.
Crack eggs into a large mixing bowl. Add crushed garlic, oregano, pepper, and cayenne pepper; whisk together.
Transfer cauliflower to mixing bowl. Add shredded Cheddar cheese and grated Parmigiano-Reggiano. Mix thoroughly.
Transfer to prepared baking dish. Tamp down and smooth out evenly.
Bake in preheated oven until deeply golden brown, about 1 hour. Allow to cool to room temperature.
Cover and refrigerate until thoroughly chilled, 3 or 4 hours.
Preheat oven to 400 degrees F.
Line a baking sheet with a silicone mat; lightly oil the mat.
Lift cauliflower out of pan. Slice in half lengthwise; slice each half crosswise into thin "fries," about 1 inch wide.
Arrange fries on the mat so they do not touch; brush lightly with olive oil.
Sprinkle with a dusting of grated Parmigiano-Reggiano cheese.
Bake in preheated oven until deeply browned, about 20 to 25 minutes.