Slice the portobellos into 1/2-inch thick strips, then cut each strip in half crosswise. Repeat with the shiitakes, keeping the two mushrooms separate.
Melt 2 tablespoons of the butter in a large skillet over medium heat. When the foam starts to subside, add the portobellos and cook, tossing once or twice, about 1 minute. Add 1/4 teaspoon salt and a generous pinch of black pepper, and cook, stirring occasionally, until browned, juicy and tender but firm, about 2 minutes. Transfer to a medium bowl.
Add 1 tablespoon of the remaining butter to the skillet, let it sizzle, then add the shiitakes and toss for about 30 seconds. Add the remaining 1 tablespoon butter, 1/4 teaspoon salt and a generous pinch of black pepper, and cook, tossing occasionally, until browned and tender but still firm, 1 to 2 minutes.
Return the portobellos to the skillet, sprinkle the flour over the mushrooms, then toss, scraping the bottom of the skillet, until the flour disappears, about 30 seconds. Stir in half the broth and all the soy sauce and mustard, scraping the bottom of the skillet until it's clean, then stir in the remaining broth. Bring the mixture to a simmer, and cook until the sauce thickens slightly, about 6 minutes. Cover, reduce the heat to low and simmer 5 minutes, stirring once or twice. Stir in the goat cheese.
Remove the skillet from the heat, and stir in the sour cream and parsley. Taste and add more salt and pepper as needed. Serve the stroganoff over buttered egg noodles.
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