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Preheat oven to 375 degrees; butter a 1 1/2-quart baking dish and set aside.
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Bring a large pot of water to a boil over high heat; generously salt water and return to a boil. Add kale and cook until tender, 4 to 5 minutes. Drain and rinse under cold running water to cool. Squeeze water from kale until completely dry. Finely chop; you should have about 2 1/2 cups chopped kale. Transfer to a large bowl and set aside.
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Place sausage in a large skillet and cook, stirring, until brown; drain fat from skillet and transfer sausage to a bowl with kale.
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Melt 1 tablespoon butter and oil in skillet over medium-high heat. Add mushrooms and cook, stirring, until lightly brown and tender, about 5 minutes. Add vinegar to deglaze and transfer mushrooms to bowl with kale and sausage.
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In a medium saucepan, melt remaining 4 tablespoons butter over medium heat. Add flour and cook, whisking, until thick and bubbly, about 3 minutes. Whisk in milk and cream. Increase heat to medium-high; cook until mixture just begins to boil. Reduce heat to medium-low and simmer, whisking often, about 3 minutes. Remove from heat and stir in 1 cup Parmesan and 1 cup Gruyere; stir in lemon zest, 2 teaspoons salt, and pepper. Transfer cheese mixture to bowl with kale mixture and mix until fully combined.
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Transfer kale mixture to prepared baking dish; top with remaining 1/2 cup Parmesan and Gruyere. Bake until heated through, about 25 minutes.
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Preheat broiler. Transfer baking dish to broiler; broil until browned and bubbly, 1 to 2 minutes. Serve dip with tortilla chips or toasted baguette.
