Butternut Squash Flan with sauce

2 1/2 lbs butternut squash, halved lengthwise and seeded 4 large eggs , plus 2 egg yolks 2 cups half-and-half 1 1/8 ounces finely grated parmigiano-reggiano[...]

1/2 cup sugar 1/4 cup water 1 1/2 cups butternut squash (cooked and mashed) 3 eggs 3 egg yolks 2 tablespoons brown sugar 1/2 teaspoon salt 1/2 teaspoon ginger[...]