Set up: Heat the oven to 200°F. Put a baking sheet inside to warm, and set a large nonstick skillet and a medium saucepan on the stove.

Put the chopped broccoli and 2 tablespoons water in a microwave-safe bowl. Cover the bowl and microwave for 2 minutes, until the broccoli is bright green and just tender.

Drain the broccoli in a colander. Spread it on a clean kitchen towel or paper towels and pat it very dry. Let it cool for 2 minutes so it does not warm the egg mixture too much.

Start the beans: Add 1 tablespoon olive oil to the medium saucepan and set it over medium heat.

Add the crushed tomatoes, drained butter beans, 1/2 teaspoon kosher salt, 1/4 teaspoon black pepper, and the red pepper flakes to the saucepan. Stir well and bring the mixture to a steady simmer.

Lower the heat to medium-low and cook the beans for 10 minutes, stirring once or twice, until the tomatoes thicken slightly and coat the beans. Keep the saucepan on low heat while you cook the fritters.

Mix the fritters: Put the dried broccoli in a large bowl. Add the grated cheddar, most of the sliced scallions, reserving a small handful of green tops for serving, the flour, the remaining 1 teaspoon kosher salt, the remaining 1/4 teaspoon black pepper, and the zest from half the lemon.

Crack the eggs into a small bowl if you have not already beaten them. Beat them with a fork until the yolks and whites are fully mixed.

Pour the beaten eggs into the broccoli bowl. Stir until no dry flour remains and the mixture holds together when you press a spoonful against the side of the bowl.

Fry the fritters: Add 2 tablespoons olive oil to the large nonstick skillet and heat it over medium heat until the oil looks loose and shiny.

Scoop four 1/4-cup mounds of the broccoli mixture into the skillet. Press each mound gently with the back of a measuring cup or spatula to make a patty about 1/2 inch thick.

Cook the fritters for 3 to 4 minutes on the first side, until the bottoms are deep golden brown and the edges look set.