Melt 2 tablespoons butter in heavy large skillet over medium-high heat. Add maitake mushrooms and sprinkle with sea salt. Sauté mushrooms until tender and beginning to brown, 3 to 4 minutes. Set aside.
Bring the vegetable broth to a simmer in a medium saucepan; keep warm. Melt remaining 1 1/2 tablespoons butter with olive oil in heavy large saucepan over medium-low heat. Add shallots, sprinkle with salt, and sauté until tender, 4 to 5 minutes. Add rice and increase heat to medium. Stir until edges of rice begin to look translucent, 3 to 4 minutes. Add white wine and Vermouth and stir until liquid is absorbed, about 1 minute. Add 3/4 cup warm broth; stir until almost all broth is absorbed, about 1 minute. Continue adding broth by 3/4 cupfuls, stirring until almost absorbed before adding more, until rice is halfway cooked, about 10 minutes. Stir in sautéed mushrooms and escarole. Continue adding broth by 3/4 cupfuls, stirring until almost all absorbed before adding more, until rice is tender but still firm to bite and risotto is creamy, about 10 minutes. Stir in 1/4 cup grated Parmesan cheese, if using. Transfer risotto to serving bowl. Pass additional Parmesan cheese alongside, if desired.