Bring a pot of water to a boil and season with Kosher salt. Add the peas and garlic clove and cook until tender, 5-7 minutes.
Drain peas, reserving about ½ cup of the cooking water, and transfer to a food processor or blender. Remove garlic clove, add butter and parmesan then blend until smooth, adding cooking water to get the consistency you want. Adjust seasoning with salt and black pepper. Transfer to a microwave-safe bowl and set aside.
Just before shrimp are finished, reheat the pea puree in 15-second intervals in the microwave until warm, then divide them among 4 plates.
Butter Poached Shrimp
In a large heavy-bottomed saucepan, melt butter over medium heat and cook until it begins to bubble, about 5 minutes. Once foam starts to appear on the surface, reduce the heat to medium-low and use a spoon to gently skim away foam and discard it. Continue gently simmering the butter and skimming the foam until the water has cooked off and no more foam appears, 15-20 minutes. The surface should be clear and glassy.
Place a mesh strainer over a large bowl, line it with a double layer of cheesecloth and pour the butter through the strainer. Discard the cheesecloth, wash and dry the saucepan, and return the warm butter to the pot.
Season the butter with salt and heat over low heat.
Add the shrimp to the butter, cooking 10-12 at a time and cook gently over low heat, stirring occasionally, until the shrimp are cooked through, 4-6 minutes.
Use a slotted spoon to place the shrimp over the pea puree, finish with a drizzle of poaching butter and sprinkle with fresh parsley.