Heat the oven to 400° F. Toss the squash with the oil, put on a baking sheet, and roast until just tender, 30 to 40 minutes. Reduce the oven temperature to 350° F.
Warm the milk, butter, sage, and a good sprinkling of salt and pepper in a small saucepan over low heat just until the butter melts. Meanwhile, butter a 1 1/2-quart or 8-inch square baking dish (glass is nice) and cut or tear the bread into bite-sized pieces — not too small.
Put the bread in the baking dish and pour the warm milk over it. Let it sit for a few minutes, occasionally submerging any pieces of bread that rise to the top. Beat the eggs briefly and stir them into the bread mixture along with the cheese and squash. Set the baking dish in a larger baking pan and pour boiling water into the pan to within about an inch of the top of the dish.
Bake for 45 to 60 minutes, or until a thin knife inserted in the center comes out clean or nearly so; the center should be just a little wobbly. Run under the broiler for about 30 seconds if you like, to brown the top a bit. Serve hot or store, covered, in the refrigerator for up to 2 days. Cover with foil and reheat in a 325° F oven for about 15 minutes; remove the foil and heat for another 5 minutes or so for a crisper crust.
Time: About 1 hour, largely unattended