Porcini-Gorgonzola Burgers with Veal Demi-Glacé

For demi-glace:

Heat oil in heavy large pot over medium-high heat. Add veal bones and sauté until deep brown on all sides, about 12 minutes. Transfer bones to bowl. Add onion, carrot, and celery to pot. Sauté until browned, about 6 minutes. Add 2 cups cold water, tomato paste, herbs, and peppercorns; bring to boil, scraping up browned bits. Add bones with any accumulated juices, then remaining 10 cups water to pot. Bring to boil. Reduce heat to medium-low and simmer uncovered until stock is reduced to 2 1/2 cups, about 4 hours. Strain into 4-cup glass measuring cup, pressing on solids to extract as much liquid as possible. Refrigerate stock uncovered 1 hour.

Spoon off fat from surface of stock. Place stock in heavy medium saucepan; add wine and simmer until reduced to 1 cup demi-glace, about 25 minutes. (Can be made 3 days ahead. Cover and chill.)

For burgers:

Bring 2 cups water and mushrooms to boil in small saucepan. Remove from heat, cover, and let stand until mushrooms soften, about 20 minutes. Using slotted spoon, transfer mushrooms to medium skillet. Slowly pour soaking liquid into skillet, leaving any sediment behind. Boil over medium-high heat until liquid is reduced to glaze, stirring occasionally, about 6 minutes. Cool mushrooms. Transfer to work surface; chop mushrooms coarsely.

Transfer mushrooms and any juices to large bowl. Add meat, salt, and pepper; mix gently. Shape mixture into six 1-inch-thick patties. Place on foil-lined baking sheet. (Can be made 1 day ahead. Cover and chill.)

Preheat broiler. Rewarm demi-glace over medium-low heat; add butter and whisk until melted. Remove from heat. Place rolls, cut side up, on baking sheet. Broil until lightly toasted, about 2 minutes. Remove rolls; maintain broiler setting. Arrange roll bottoms on plates. Top with arugula. Heat 2 large broilerproof skillets over medium-high heat; brush with oil. Add 3 burgers to each skillet. Cook to desired doneness, about 5 minutes per side for medium-rare. Top each burger with slice of cheese. Broil burgers until cheese begins to melt, about 1 minute. Place burgers on roll bottoms; spoon some demi-glace over. Cover with roll tops. Serve, passing remaining demi-glace separately.

Dried porcini mushrooms are available at Italian markets, specialty foods stores, and many supermarkets