1. Heat the oven to 350 degrees. Have 4 shallow ramekins and a baking sheet at hand. (The sheet is optional, although it is handy for handling the ramekins.) If using a shallot, mince it now. Grate the cheese if needed.
2. Place ½ teaspoon of the butter into each ramekin. Heat on low (either in the microwave or on the stove top) until the ramekins are hot and the butter has melted. Swirl to coat evenly.
3. Crack open 2 eggs, letting them fall into a medium bowl. Use a large perforated spoon to transfer them to one of the ramekins, letting any wispy drips of egg white strain away as you work. Repeat to use all the eggs.
4. Arrange the ramekins on the baking sheet. Pour 1 tablespoon of the cream or half-and-half into each ramekin. Sprinkle in a teaspoon or 2 of the cheese and some of the shallot, if using. Bake for about 10 minutes, starting to check for doneness at 8 minutes. The whites should be just set, with maybe a bit that has not yet become opaque, and the yolks should look as if they could break and run easily. If you wish to finish the eggs under the broiler (to brown the cheese), remove the eggs from the oven after 5 to 7 minutes, then position the top oven rack 4 to 6 inches from the broiling element. Broil the eggs just until the cheese develops a little color.
5. Season lightly with salt and pepper, and garnish with a few tarragon leaves just before serving.
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