To make the beurre blanc, add the minced shallots, vinegar, and wine to a saucepan. Simmer over medium heat until it is reduced by more than half, about 5 minutes.
Slowly add the cream and stir to incorporate.
Add the butter chunks to the pan all at once. Turn the heat to high and whisk while moving the pan until it has completely melted. Season with salt and pepper to taste.
Strain out the shallots with a fine mesh sieve.
Make The Pesto Pea Purée:
To make the pesto pea puree, boil the water in a saucepan, then add the garlic clove and frozen peas. Cook the peas until tender, about 3 minutes. Strain the peas, but reserve some of the liquid. Remove garlic clove. Add the peas to a food processor or blender with the pesto, and season with salt and pepper to taste. Pulse until you get a consistency you like. Use a little of the cooking water to thin if necessary. Remove from the food processor and mix in the Parmesan.
Sear The Scallops:
Pat the scallops dry with a paper towel, then season with salt and pepper. Heat a sauté pan over high heat. Add butter and oil. When the butter is melted, place scallops in the pan.
Leave them alone for about a minute and a half before flipping each scallop carefully. They will develop a golden crust.
Add pesto pea puree to a plate. Arrange seared scallops on top, then pour beurre blanc on top.