Melt 2 tablespoons of butter in a heavy large skillet over medium-high heat.

Add shiitake and oyster mushrooms and thyme; saute for 5 minutes.

Add 2 tablespoons of chopped shallots; saute until mushrooms are tender, about 3 minutes.

Add Madeira wine and simmer until almost all liquid evaporates, about 2 minutes.

Add whipping cream and simmer until slightly thickened, about 2 minutes.

Set the mushroom-Madeira sauce aside.

Cook green beans in a large pot of boiling salted water until just tender, about 5 minutes.

Drain the green beans.

Transfer the green beans to a bowl of ice water to cool.

Drain the green beans.

Pour vegetable oil into a large deep saucepan to reach a depth of 4 inches.

Heat the vegetable oil to 350°F.

Place 1/4 of thinly sliced leeks in a small metal strainer.

Lower the strainer into the hot oil and fry until golden, about 40 seconds.

Lift the strainer from the oil.

Drain the fried leeks on paper towels.

Repeat the frying process with the remaining leeks in 3 more batches.

Season the fried leeks with salt.

Melt 1 tablespoon of butter in a heavy large skillet over medium heat.

Add the cooked green beans and the remaining 1 tablespoon of chopped shallots; toss to heat through.

Season the green beans with salt and pepper.

Place the green beans on a platter.

Bring the mushroom-Madeira sauce to a simmer.

Spoon the sauce over the green beans.

Sprinkle with the fried leeks.