Heat oven to 350ºF.
Heat the oil in a skillet set over medium heat. Add the onion and cook 3-4 minutes, or until softened. Add the broccoli, jalapeno, and water. Cook an additional 3-4 minutes, or until the broccoli is soft and the water is evaporated. Remove from heat and let cool completely, about 10 minutes. Stir in the eggs, cheese, and cilantro.
To prepare the phyllo, lay one sheet out in front of you on a clean surface, with the short end facing you. (Cover the remaining sheets with a damp towel to keep them from drying out.) Lightly brush the phyllo sheet with olive oil. Place a second sheet directly on top of the first sheet. Cut the phyllo into three long strips.
Place one heaping tablespoon of filling on the end of each phyllo strip that’s closest to you. Fold the corner up around the filling to make a small triangle, then keep flipping/rolling the triangle until you get to the end (like you’re folding a paper football or a flag.) Brush each triangle with oil and arrange on a baking sheet in a single layer. Repeat with remaining phyllo sheets until you have no filling left.
Bake for 25 minutes, or until crisp and light golden brown. Serve with salsa for dipping.