In a large heavy skillet over medium heat, add three tablespoons of butter to the olive oil and heat until melted.
Add the mushrooms, green onions, and thyme, and cook, stirring occasionally, until liquid has been released and skillet becomes almost dry, 5 to 8 minutes. Stir in salt, pepper, and sherry and cook until skillet is almost dry, about 10 minutes more. Remove from heat, and let cool.
Combine mushroom mixture with reserved walnuts, parsley, lemon juice, Tabasco, and cream cheese in a blender and pulse until the mixture is smooth.
Season with salt and pepper to taste.
Spread mixture into a shallow plate or bowl and refrigerate for at least 4 hours or overnight.
Just before serving, melt remaining 1 tablespoon of butter in a medium skillet and cook halved mushrooms just until tender, about 5 minutes. Season with salt and pepper.
Remove the pate from the refrigerator and top with the sauteed mushrooms and chopped parsley. Offer chopped hard boiled egg as an alternative.
Serve cool or at room temperature with toasted baguette slices, or any plain cracker.