Preheat oven to 200°F. Separate broccoli florets from the biggest stem(s), cutting into 1-inch chunks. Slice stems into 1/4-inch lengths; this should equal about 3 cups of chopped broccoli. To steam broccoli until tender but not mushy, bring 1/2 inch of water to a boil in a medium saucepan, add broccoli, cover, and simmer for 5–6 minutes. Drain broccoli, then set aside to cool slightly in a bowl.
In bottom of a large bowl, lightly beat 1 egg and 1 tablespoon water. Add flour, cheese, garlic, mustard, salt, and pepper flakes. Use a potato masher or large fork to mash the broccoli just a bit. Keep the bits recognizable, but small enough (1/4- to 1/2-inch chunks) that you can press a mound of the batter into a fritter in the pan. Once mashed, fold the ingredients together with a spoon.
Heat a large cast-iron skillet over medium-high heat. Once hot, add 2 to 3 tablespoons of oil. When the oil is hot, drop a 2-tablespoon mound of the batter into the pan and flatten it slightly with a spoon or spatula. Repeat with additional batter, leaving a couple inches between each fritter. Once brown underneath, about 2–3 minutes, flip each fritter and cook on the other side until equally golden, about another 1–2 minutes.
Transfer briefly to paper towels to drain, then to a baking sheet, and place in oven to keep warm while cooking remaining fritters. Serve with fennel yogurt.