Preheat oven to 375 degrees. Line baking sheet with parchment paper.
Steam broccoli and carrots until just fork tender, about 5 minutes.
Meanwhile sauté onion in olive oil until translucent. Add in cooked chicken, cream cheese, and shredded cheddar. Stir to combine then gently fold in steamed broccoli and carrots. Season to taste.
Place puff pastry on parchment lined baking sheet. Cut about 12 strips into the outer thirds of the pastry, leaving center uncut.
Mound filling down the center of the pastry. Crisscross strips of pastry over filling, overlapping as you go. Brush top of Stromboli with beaten egg.
Bake for 30 minutes or until golden brown. Allow to rest for 5 minutes before slicing.