How to Make It
In a small heatproof bowl, cover the dried porcini mushrooms with the boiling water. Let stand until the mushrooms have softened, about 15 minutes.
Meanwhile, in a medium bowl, toss the white mushrooms with 2 teaspoons of the lemon juice. In a large skillet, melt 1 tablespoon of butter in 1 tablespoon of the olive oil until sizzling. Add the white mushrooms and season with salt and pepper. Cover and cook over moderate heat until the liquid released from the mushrooms has evaporated, about 6 minutes. Uncover and cook, stirring occasionally, until the mushrooms are browned, about 4 minutes. Transfer to a plate.
In the same skillet, melt 2 tablespoons of the butter in the remaining 1 tablespoon of oil. Add the chanterelles and season with salt and pepper. Cover and cook over moderate heat until the mushrooms are swimming in liquid, about 5 minutes. Uncover and cook over moderately high heat until the liquid has evaporated, about 3 minutes. Continue to cook, stirring occasionally, until the chanterelles are browned, about 8 minutes. Add the shallots and cook over moderate heat, stirring a few times, until softened, about 4 minutes.
Using a slotted spoon, transfer the porcini to a work surface. Coarsely chop the porcini and add them to the chanterelles along with their soaking liquid, stopping before you reach the grit. Stir in the mushroom broth and the white mushrooms and simmer over moderate heat until the porcini are soft, about 4 minutes.
In a small bowl, blend the 1/2 tablespoon of softened butter with the flour to make a smooth paste. Stir in 1/3 cup of the ragout liquid to dissolve the paste, then whisk the mixture into the ragout. Add the minced garlic, chopped thyme and the remaining 1 teaspoon of lemon juice and simmer over low heat, stirring occasionally, until the ragout is thickened and glossy, about 4 minutes. Season the mushroom ragout with salt and pepper and serve with soft polenta.