Fill the bottom of a large skillet with 1/2 inch of canola oil and heat over medium high heat until 360 degrees F. Make sure the oil is no more than halfway up the side of the skillet. Oil is hot enough when the end of a wooden spoon is submerged and small bubbles appear around it quickly.
Working in small batches, carefully add the sliced shallots and fry until golden and crispy. Remove with slotted spoon and drain on paper towel. Season with salt and pepper if desired. Set aside and reserve.
Preheat oven to 425 degrees F.
Separately toss the green beans and mushrooms with 1/2 Tbsp. each of the oil from the shallots. Season with salt and pepper if desired.
Place each vegetable on a sheet pan. Mushrooms can release a lot of liquids during roasting and you don't want to change the texture of the green beans.
Place both halves of the garlic on a small piece of aluminum foil and sprinkle with 1 tsp. of the shallot oil. Lightly crumble foil up to seal package and place on sheet pan with green beans or directly on rack in oven.
Roast vegetables for about 30 to 45 minutes until lightly caramelized and tender. The mushrooms will likely be done first. Remove and set aside as they are done.
Begin preparing the sauce by gently heating the half and half, thyme, nutmeg, white pepper and soy sauce over low heat in sauce pan.
Remove roasted garlic from foil package and squeeze the cloves into the bowl of a food processor along with the 1/2 cup reserved roasted mushrooms. Pour in about 1/2 cup of the half and half mixture and pulse until smooth. Pour mixture back into the sauce pan with the half and half mixture and whisk to combine.
While the mixture is heating beat the egg yolk in a small bowl. Slowly whisk the half and half mixture into the egg yolk a tablespoon at a time until the egg yolk is warmed. It may take two or three additions. Keep whisking and pour back into the sauce pan with the half and half mixture.
Reduce heat to low and continue whisking until mixture comes to a simmer. Simmer until mixture is reduced by about half, whisking occasionally. The sauce should be velvety, thickened and coat the back of a spoon.
Pour through a strainer to remove any cooked egg yolk and mushroom pieces. Stir in the sherry and cook for an additional 2 or 3 minutes.
To assemble, arrange roasted vegetables on a platter and spoon sauce across the top. Garnish with fried shallots.