Preheat air fryer to 325°. Place one sheet of phyllo dough on a work surface; brush with butter. Layer with four additional phyllo sheets, brushing each layer. (Keep remaining phyllo covered with plastic wrap and a damp towel to keep it from drying out.) Cut layered sheets crosswise in half, then lengthwise in half.
Place each stack in a greased 4-oz. ramekin. Fill each with a piece of cream cheese. Carefully break an egg into each cup. Sprinkle with salt and pepper; top with ham, cheese, bread crumbs and chives. Bring phyllo together above filling; pinch to seal and form bundles.
Place ramekins in air-fryer basket; brush with remaining butter. Cook until golden brown, 10-12 minutes. Serve warm.