Bring a large pot of salted water to a boil. Add broccoli florets and cook for 3 minutes until bright green and just tender. Drain immediately and plunge into ice water to stop cooking. Drain well, then spread on a clean kitchen towel and press out as much moisture as possible. Transfer to a cutting board and chop finely - you want pieces about the size of rice grains for patties that hold together.

In a large bowl, combine the chopped broccoli, 1 cup shredded cheddar, beaten eggs, panko, flour, garlic powder, onion powder, salt, and pepper. Mix thoroughly with your hands until the mixture clumps together when squeezed. If it feels too wet, add another tablespoon of panko; if too dry, add a teaspoon of water.

Divide mixture into 8 equal portions and shape into 3-inch patties about 3/4 inch thick. Press them firmly so they hold together - compact patties fry better than loose ones. Place on a plate and refrigerate for 10 minutes while you heat the pan; this helps them set.

Heat olive oil in a large nonstick skillet over medium heat. When the oil shimmers, add patties in batches - do not crowd the pan. Cook for 3-4 minutes per side until deeply golden and crisp. In the last minute of cooking, sprinkle remaining 1/2 cup cheese over the patties and cover briefly to melt. Transfer to a wire rack or paper towel-lined plate. Repeat with remaining patties, adding more oil if needed.