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Preheat the oven to 400 degrees.
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Wash the potatoes and prick each a few times with a fork. Bake the potatoes until they start to soften but are definitely not completely done, about 20 minutes. Let cool about half an hour. (This can be done a day ahead of time.) Leave the oven on.
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Grate the semi-baked potatoes with a hand-held grater - peel them first if you like, although it’s not necessary. In a medium bowl put the grated potatoes, cheese, salt, pepper, and butter; toss well to combine.
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Spray the mini muffin tin with nonstick spray. Using about 2 tablespoons of the potato mixture for each mini muffin cup, press the potato mixture into the muffin cups, making a well in the center and pushing the potatoes up and over the sides of the cup (they will shrink in the oven).
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Bake the hash brown “baskets” 20-30 minutes, until the undersides are deep golden brown and the tops are at least light golden and starting to crisp around the edges. Again, leave the oven on.
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Remove the baskets from the oven. Put a small dollop of pesto into each cup, then crack a quail egg on top of the pesto. Put the baskets back in the oven for exactly 5 minutes; you want the whites of the quail eggs cooked but the yolks still very runny.
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Use an offset spatula to put the hash brown baskets on a serving plate. Sprinkle with salt and serve immediately.