How to Make It
Preheat the oven to 350°. In a large skillet, melt 2 tablespoons of the butter. Add the mushrooms and season with salt and pepper. Cover and cook over moderate heat, stirring a few times, until the mushrooms release their liquid, about 8 minutes. Uncover and cook, stirring occasionally, until the liquid has evaporated and the mushrooms are browned, about 8 minutes longer.
In another large skillet, melt 2 tablespoons of the butter. Add the onion and cook over moderate heat, stirring occasionally, until softened, about 7 minutes. Stir in the flour until incorporated, then gradually stir in the broth until smooth. Bring to a boil over moderately high heat and cook, stirring often, until thickened, about 2 minutes. Stir in the mushrooms, thyme and lemon zest. Remove from the heat and season with salt and pepper. Let cool completely.
In a small saucepan, melt the remaining 1 stick of butter. Line a large rimmed baking sheet with parchment paper. Place 1 phyllo sheet on the parchment and brush lightly with some of the melted butter. Top with the remaining phyllo, buttering each sheet as you go; reserve about 1 1/2 tablespoons of the butter. Spoon the cooled mushroom filling onto a long side of the phyllo and carefully roll it up like a log, folding in the ends as you go. Shift the strudel to the center of the baking sheet, seam side down. Brush all over with the reserved melted butter. Bake for about 45 minutes, until the strudel is golden brown and crisp. Let rest for about 10 minutes, then slice and serve.