In a small saucepan, heat butter and oil. Add shallot and saute until soft and fragrant, 3-5 minutes. Add herbs and salt; remove from heat and stir to combine. The mixture should be somewhat coarse and just a little buttery.
Preheat the oven to 375. Place a small pat of butter (about 1 teaspoon) in the bottom of each of 3 ramekins. Place in oven until butter is melted.
Remove ramekins from oven, add 1 teaspoon of cream to each, and crack 2 eggs into each ramekin without breaking the yolk. Sprinkle the herb mixture over the top of each ramekin and return to the oven. Bake for about 7 minutes.
Turn up the heat to the broil setting and let the eggs broil for another 1-2 minutes. Remove from oven when egg whites are just set and yolks are still soft. Let stand for 3-5 minutes before serving. Season with salt and pepper.
