To prepare the galette rounds: In bowl whisk together milk, butter, yolks, and whole egg. In another bowl whisk together flour, salt, and chives and stir into milk mixture until just combined. On a lightly floured surface with floured hands knead dough about 8 times, or until just smooth. Wrap dough in plastic wrap and chill for 1 hour.
Preheat oven to 450 degrees F.
Divide dough into 4 pieces. On a lightly floured surface with a floured rolling pin roll out each piece into an 8-inch round. Transfer rounds to 2 baking sheets and crimp edges decoratively. Chill dough 10 minutes and bake in middle and lower-third of oven about 5 minutes, or until golden brown. Transfer galettes to racks and cool completely. Galettes may be made 1 day ahead and kept in a sealable plastic bag at room temperature.
To prepare the onions: In a skillet cook onions in oil and butter with salt and pepper, to taste, over moderately low heat 15 minutes, or until golden. Transfer onions to a bowl to cool.
Have ready a large bowl of ice and cold water. Cut asparagus crosswise into 1/2-inch pieces and cook in a large saucepan of boiling salted water 3 to 5 minutes, or until just tender. Drain asparagus in a colander and transfer to bowl of ice and cold water to stop cooking. Lift asparagus out of water and pat dry.
Preheat oven to 400 degrees F.
Spread onions evenly onto galette rounds and top with prosciutto, asparagus, and goat cheese. Bake galettes on a baking sheet in middle of oven about 15 minutes, or until tops are lightly browned. Transfer galettes to a rack and cool.
Serve galettes, cut into wedges, at room temperature. Garnish with chopped chives.
Recipe courtesy of Gourmet Magazine