Prep your ingredients: wash broccoli and divide into florets. Peel potatoes and cut into 1 inch cubes.
Bring water to a boil in a medium sized sauce pan. Add broccoli; potatoes and salt. Cook until fork tender, about 12-15 minutes. Drain and transfer to a bowl.
Mash the broccoli and potatoes with a potato masher. Let cool for about 10 minutes before adding remaining ingredients.
Preheat oven to 400 degrees F and line a baking sheet with parchment paper and coat with 1 tbsp. of olive oil.
Add grated cheese; ½ cup of breadcrumbs; parsley; garlic and salt and pepper to the broccoli potato mixture. Combine all ingredients well. Taste and adjust seasoning, if necessary, before stirring in the egg.
Shape into 12 patties, dredge both sides in the remaining ¼ cup of breadcrumbs and place on the prepared baking sheet. Drizzle more olive oil over the top.
Bake for 30 minutes, flipping the patties halfway through baking. Serve hot.
Notes
Vary the flavor by adding other ingredients such as chives; green onions; thyme; or dill.
Add a cube of mozzarella cheese to the center of each patty for extra cheesiness.
Keep leftovers refrigerated for up to 3 days. Reheat in the oven.
Make ahead: shape the patties up to one day in advance and keep refrigerated until ready to bake.
Freeze cooled, baked patties in freezer proof containers. Defrost for a few hours or overnight in the refrigerator and bake until heated through.