Trim, halve, and thinly slice fennel. Dice onion, chop red pepper, and mince garlic. Coarsely chop basil. Shred chicken into bite-size pieces.

Heat oven to 400°F. In a large skillet heat oil over medium-high heat. Add onion, and cook 2 minutes, or until slightly softened. Add fennel and cook, stirring occasionally, until vegetables are lightly golden. Stir in pepper and garlic; cook until softened, about 4 minutes. Transfer vegetable mixture to a large bowl. Add chicken, basil, and chicken broth. Stir to combine.

To assemble pie, first melt butter. Lightly butter bottom and sides of a round 9-inch springform pan. Place 2 sheets of phyllo in bottom of pan, allowing sheets to overhang on one side. Brush phyllo inside pan generously with butter. Layer remaining sheets in pan, 2 at a time, with sheets overhanging sides, in a spoke-wheel fashion, buttering between the doubled sheets of phyllo.

Sprinkle bread crumbs in bottom of phyllo-lined pan. Crumble half the feta cheese over bread crumbs. Spoon chicken-vegetable mixture into pan; crumble remaining feta cheese on top. Working quickly so phyllo won’t dry, fold in each layer of overhanging dough to cover pie filling, buttering between doubled sheets. Bake 25 to 30 minutes, or until golden brown. Let rest 5 minutes in pan before serving.