Preheat oven to 375 degrees F. In an 8-inch cast-iron skillet or other oven-going skillet heat 2 teaspoons of the oil over medium-high heat. Add onion, garlic, and crushed red pepper; cook for 3 to 4 minutes or until onion is tender, stirring occasionally. Stir in whipping cream; bring to boiling. Remove from heat.
In a 4- to 5-quart Dutch oven heat another 2 teaspoons of the oil over medium-high heat. Add spinach; cook and stir about 3 minutes or just until wilted. Drain spinach in a colander, pressing gently to remove excess liquid. Return spinach to Dutch oven. Stir in chicken, cheese, parsley, and the 1 tablespoon dill weed. Add spinach mixture to onion mixture in skillet; stir to combine.
Unfold phyllo dough; remove one sheet. (As you work, keep the remaining phyllo covered with plastic wrap to prevent it from drying out.) Brush phyllo sheet with some of the melted butter. Fold in half to make a 9x7-inch rectangle. Place on top of spinach mixture in skillet. Repeat with the remaining phyllo, brushing each sheet with butter and rotating slightly when placing on spinach mixture to stagger corners. (Don't worry if phyllo sheets tear a little.) Brush with any remaining butter. Using a sharp knife, cut a few slits in phyllo.
Bake about 15 minutes or until top is golden. If desired, garnish with additional dill weed.