Lay the puff pastry over the skillet for a pretty look. Use any brand; the weight may vary, but either way you'll have enough to work with.
Place a rack in upper third of oven; preheat to 425°F. Heat oil in an 8" cast-iron or other heavy ovenproof skillet over medium-high heat. Add onions; cook, stirring occasionally, until beginning to brown, about 4 minutes.
Reduce heat to medium-low. Add garlic and sage to skillet and cook, stirring occasionally, until garlic begins to brown, about 2 minutes.
Stir in broth, 1/2-cupful at a time, then add squash. Bring to a boil, reduce heat, and simmer until squash is just softened and broth is thickened, 8-10 minutes. Add chicken to skillet, stir, and season with salt and pepper.
Unfold pastry and smooth any creases; place over skillet, allowing corners to hang over sides. Whisk egg and 1 teaspoon water in a small bowl. Brush pastry with egg wash; cut four 1" slits in top to vent.
Bake pot pie until pastry is beginning to brown, 15-20 minutes. Reduce oven temperature to 375°F and bake until pastry is deep golden brown and crisp, 15-20 minutes longer. Let cool for 10 minutes before serving.