Clean shrimp, separating heads from tails. With a pairing knife, make an incision on the underside, open up the shell and use your fingers to remove the meat, placing it in a bowl. Use the heads to make a fumet (a reduced broth), using about 6 cups of water and 2 teaspoons salt.
Bring to a boil and simmer over low heat for at least 20 minutes. Keep warm. Clean mushrooms by cutting off the tips of the stems and clean, if necessary, with a wet dishcloth. Cut into thin slices.
Heat oil in a pan over medium heat. Sauté peeled garlic, but do not let it brown. When golden, add shrimp and mushrooms and sauté for 2 more minutes. Add rice and toast it for a couple of minutes, or until it becomes translucent.
Add wine and allow to reduce until almost dry. Add a couple ladlefuls of hot strained fumet. Lower the heat and continue adding fumet, a ladleful at a time, as it is absorbed by the rice.
After about 12 minutes, add washed, chopped arugula. After 5 more minutes, season with salt and pepper. Remove from the heat and stir in butter and cognac.
If the rice looks too dry, add more fumet or warm water before serving, stir and serve immediately.