Butternut Squash Flan with Parmesan Sage Sauce

Put oven rack in middle position and preheat oven to 400 °F. Roast squash, cut sides down, in a lightly oiled shallow baking pan until neck is tender, about 1 hour. Remove from oven and reduce oven temperature to 325 °F. Cool squash to warm. Butter an 8- by 2-inch round cake pan. Line bottom with a round of parchment or wax paper and butter paper. Scoop flesh from squash, discarding skin, and purée in a food processor until smooth, about 45 seconds. Put 2 cups purée in a bowl, then whisk in whole eggs, 1 cup half-and-half, 2 tablespoons cheese, 1 1/4 teaspoons salt, and 1/2 teaspoon black pepper until combined. Pour squash mixture into cake pan and bake in a water bath until just set and a wooden pick or skewer comes out almost clean, 45 minutes to 1 hour. Remove from water bath and cool flan in pan on a rack 15 minutes. Invert a flat plate over flan, then invert flan onto plate and carefully remove parchment. Meanwhile, bring whole sage leaves and remaining cup half-and-half just to a simmer in a 1- to 2-quart heavy saucepan, then remove from heat and let steep, covered, 10 minutes. Remove sage, carefully squeezing leaves to extract liquid, and discard leaves. Whisk yolks into half-and-half and cook over moderately low heat, stirring constantly with a wooden spoon, until sauce is thick enough to coat back of spoon and registers 170 to 175 °F on thermometer, about 4 minutes (do not let boil). Immediately pour sauce through a fine-mesh sieve into a bowl, then add remaining 1/2 cup cheese and chopped sage, stirring until cheese is melted. Stir in 1/4 teaspoon salt and 1/2 teaspoon pepper. Cut flan into wedges with a thin knife, wiping knife clean after each slice. Serve flan with sauce.