In a large saute pan over medium-high heat, add the oil. Add the garlic and saute for a few seconds. Add the mushrooms, a pinch of salt and pepper, chives, and sherry vinegar. Cook until softened and slightly crisp, about 3 minutes. Remove the mushroom mixture from the pan and add to a food processor. Pulse a few times to chop finely chop the mushrooms and reserve.
In a large bowl, add eggs, cream, and salt and pepper, to taste. Beat well with a whisk until evenly incorporated.
Line the muffin tins with the tart shells and divide the mushroom mixture among the shells. Cover with the egg custard and top with the cheese. Bake until the custard sets, about 22 minutes. Remove from the oven to a serving tray and serve.