Clean mushrooms and trim stems. If the mushrooms are large, halve or quarter them. Small mushrooms can be left whole.
Heat butter and oil in a large skillet that can hold the mushrooms in a single layer. Add the mushrooms cap-side down and cook over medium high heat for 3-4 minutes, without stirring, to caramelize.
Toss the mushrooms and lower the heat to medium. Season with salt and pepper. Continue cooking until the mushrooms are tender, brown, and any released liquid is evaporated, about 5-6 minutes longer.
Add the minced garlic and cook until fragrant, about 1 minute. Deglaze the pan with 2 tablespoons of cognac and lemon juice, scraping any brown bits from the bottom of the pan. Toss the mushrooms again and cook until the cognac and lemon juice are reduced, about 2 minutes.
Add the fresh herbs to the pan and season to taste with additional salt and pepper. Just before plating, stir in the remaining 1 teaspoon of cognac.