1Set a heavy medium-size saucepan with tightfitting lid over medium-low heat. When the pan is hot, add 2 tablespoons of the butter. When the butter starts to foam, add the onion, season with salt and pepper, cover, and cook over low heat for 5 to 10 minutes, until the onion is translucent and soft but not colored. Remove from heat; keep warm.
2Meanwhile, heat a large frying pan over medium-high heat until hot, about 2 minutes. Add 1 tablespoon of the butter and cook until foaming. Add half of the mushrooms and season with salt, pepper, and 1/2 teaspoon lemon juice. Sauté, stirring occasionally, until mushrooms are browned on both sides, about 2 to 3 minutes. Transfer to a medium bowl. Repeat with remaining butter and mushrooms.
3Add mushrooms and cream to the cooked onions and bring to a simmer over medium heat until the flavors meld, about 2 to 3 minutes. Add parsley and chives, if using, and stir to combine. Taste and add salt, pepper, or lemon juice as needed. The sauce can be refrigerated for up to 5 days or frozen for up to a month.