Polenta with Green Beans, Mushrooms, Peas, and Leeks
['3/4 pound green beans, trimmed', '1 cup fresh shelled peas (12 to 14 ounces in pods)', '4 cups whole milk', '2 1/2 cups vegetable broth', '1 3/4 cups polenta (coarse cornmeal)*', '2 cups thinly sliced leeks (white and pale green parts only; about 2 medium)', '1 1/2 cups dry white vermouth', '1/4 cup (1/2 stick) chilled butter, cut into 1/2-inch cubes', '3 tablespoons heavy whipping cream', '3 tablespoons olive oil', '1 pound assorted wild mushrooms (such as crimini, small portobello, and shiitake), stemmed, caps cut into wedges', '3 large shallots, sliced (about 3/4 cup)', '2 tablespoons chopped fresh Italian parsley, divided', '2 teaspoons chopped fresh thyme']
Blanch green beans 1 minute in boiling water. Add peas and cook until both beans and peas are crisp-tender, about 3 minutes. Transfer to bowl of ice water to cool. Drain. Cut beans on diagonal into 1-inch pieces. Set beans and peas aside.
Bring milk and broth to boil in large saucepan over high heat. Reduce heat to medium. Gradually whisk in polenta. Reduce heat to low. Cook until polenta is very thick, whisking almost constantly, about 10 minutes. Season with salt and pepper. Remove from heat; cover to keep warm.
Meanwhile, combine leeks and vermouth in medium saucepan. Boil over medium-high heat until reduced to 1/3 cup, 8 minutes. Remove from heat. Gradually whisk in butter, allowing each addition to melt before adding next. Add cream and whisk over very low heat to blend. Season with salt and pepper. Set aside.
Heat oil in large skillet over medium-high heat. Add mushrooms and sauté until almost tender, 5 minutes. Stir in shallots, 1 tablespoon parsley, and thyme. Sauté until mushrooms are very tender, about 5 minutes. Add beans and peas, tossing to coat.
Rewarm polenta and spoon into large shallow bowl. Top with green bean mixture and remaining 1 tablespoon parsley. Rewarm leeks over low heat, whisking constantly; spoon evenly over polenta.
Sold at some supermarkets and at natural foods stores and Italian markets. If unavailable, substitute an equal amount of regular yellow cornmeal and cook about half as long.
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