In a blender, puree the parsley leaves and stems, lemon zest and juice, and eggs until only specks of parsley remain.
Pour into a bowl.
Add milk to the blender, swirl it around to incorporate any remaining parsley puree.
Mix the flour into the parsley puree, then mix in the milk.
Season the batter with 1/2 tsp.
salt and 1/4 tsp.
pepper.
Let it stand at room temperature for 15 minutes.
Meanwhile, bring a large pot of water to a boil.
Prepare a large bowl of ice water.
Reduce heat under the pot so the water gently simmers.
Working in 3 batches (about 3/4 cup of batter at a time), hold a colander, or a potato ricer fitted with the disk with the largest holes, over the water and, using a silicone spatula, push the batter through the colander so it falls into the water.
Cook the spaetzle until they rise to the top, about 2 minutes; using a slotted spoon, transfer to the ice water.
Repeat with the remaining batter.
When all the spaetzle are cool, drain and transfer to a medium bowl.
Drizzle with EVOO; toss until coated.
(The spaetzle can be covered and chilled for up to 3 days.
)
In a large skillet, melt the butter with the olive oil over medium-high heat.
Add the mushroom; cook, stirring often, until browned, about 5 minutes.
Add the garlic; cook, stirring often, until aromatic, about 1 minute more.
Add the spaetzle; cook, stirring often, until lightly browned, about 8 minutes.
Top with chopped parsley and cheese.