Heat one tablespoon of olive oil in a large skillet over medium-high heat. Add potatoes and garlic and sauté for about 10 minutes, stirring every now and then. Add broccoli florets, salt, pepper and cayenne pepper (if using) and give a good stir. Add vegetable broth, cover with a lid, and cook for 20 minutes, or until the vegetables are cooked through and had softened, stirring every now and then. If at any point the liquid has evaporated and the vegetables start to stick to the pan, add ¼ cup of water. Remove garlic and transfer vegetables to a large bowl and puree/mash using either a potato masher or an immersion blender (you can also use a food processor or a blender to do this.) Let cool for 5 minutes then add grated Parmesan cheese, breadcrumbs, and egg and mix well. The mixture should reach a firm consistency and hold up as patties, if not add more breadcrumbs, ¼ cup at a time, until the mixture holds together. Transfer mixture to the refrigerator for at least 20 minutes to allow the breadcrumbs to suck up some of the moisture. Take the bowl out the refrigerator, scoop ¼ cup of mixture and with dampened hands form into 12 patties (you can make larger ones if you prefer.) Heat remaining tablespoon of oil in a skillet over medium-high heat. When sizzling add patties in batches and cook for about 4 minutes on each side until browned. Serve warm!