Bring a 4- to 6-quart heavy pot of salted water to a bare simmer. Fill a large bowl with very cold water and set aside.

In a second bowl of cold water, wash leeks, then lift into a sieve to drain. Transfer leeks to a large skillet; add 1 1/2   cups fresh water, 1 tablespoon butter and generous pinch salt. Bring mixture to a simmer and cook for 10 minutes.

Transfer mixture to a blender; add 1 tablespoon of the remaining butter and purée until smooth. Transfer mixture to a small saucepan and stir in parsley.

In a large bowl, whisk together “00” flour, corn flour and 3⁄4 teaspoon salt. Add eggs and milk; with a spoon, stir together until smooth. Working over barely simmering water, force about a third of batter through spaetzle maker, food mill, or colander. As spaetzle float to surface (after about 1 minute), transfer to bowl of cold water with a mesh skimmer or sieve, then into a large bowl. Repeat with remaining batter. Drizzle drained spaetzle with oil.

Heat leek purée over medium heat to warm through. Cover to keep warm.

Trim mushrooms; keeping caps and stems intact, cut into 1/4-inch-thick slices. Melt remaining tablespoon butter in a large skillet over medium-high heat. Add mushroom slices in a single layer; add thyme. Cook until mushrooms are golden on both sides. Add spaetzle, toss to combine and cook just to warm through.

Heat 1 teaspoon oil in a small nonstick skillet over medium-high heat. Add shrimp and cook until opaque, about 1 minute per side; season with salt and pepper. Divide leek purée among 4 serving plates; top with spaetzle and shrimp.

Serve immediately.