Place the broccoli and cauliflower florets in a food processor and pulse, stopping to scrape down the sides with a wooden spoon as needed. The mixture should look evenly crumbly with tiny pieces, not wet vegetable puree.

Heat the oven to 350 °F, then wrap the vegetable crumbs in a thin kitchen cloth or paper towel and squeeze out liquid in 4 to 5 batches. Transfer the drier crumbs to a large bowl when they feel loose and no longer drip moisture.

Stir the Parmesan, egg, salt, black pepper, garlic powder, and oregano into the vegetables until evenly combined. The mixture should clump when pressed rather than spreading like a watery batter.

Line a 9x13 sheet pan with parchment paper and press the mixture evenly into the corners and along the edges. Keep the layer level so the middle firms at the same pace as the borders.

Bake for 15 minutes, scatter the cheddar over the surface, and return the pan to the oven for 10 minutes. The cheddar should melt fully and the parchment-side edges should show golden-brown caramelized spots.

Let the bread cool for 5 minutes, then lift it from the pan using the parchment and cut it into 12 rectangles. The center should feel resilient and set before slicing, with cheese that no longer pulls apart in soft strands.