Preheat oven to 425F.
Heat a medium-size skillet over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Once hot, add the onions and garlic, and cook until soft, about 6-7 minutes. Remove the pan from heat, spoon contents into a medium-size mixing bowl and let cool. Once the onions and garlic are cool enough to handle, add the chicken to the bowl with the sun-dried tomatoes, basil, oregano, fennel, chili flakes, cheese, salt and freshly ground black pepper. Mix to combine.
Place a piece of phyllo on a nonstick cookie sheet and brush with the melted butter. Top with a second sheet of phyllo and brush with butter again. Divide the chicken mixture in half and form a one-inch thick log the length of the long side of the phyllo dough. Roll the chicken log up in the phyllo tightly.
Cut the chicken-filled phyllo into 3 "cigars" and brush with a more butter.
Repeat with the remaining 2 sheets of phyllo and the rest of the chicken mixture to make a total of two chicken and phyllo roll-ups (6 "cigars"). Bake for about 12-15 minutes until golden brown.