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Coarsely grate the ginger, and place into a cheesecloth. Squeeze the ginger into a medium bowl to yield 1/4 cup juice, and then whisk in the yogurt. Season with salt and white pepper, and continue whisking until it reaches the consistency of paint. Transfer to a squeeze bottle.
Pour the liquid nitrogen into the hard plastic 4-quart container, and then drip the ginger yogurt into the nitrogen to form pearls. Transfer the pearls to a small bowl using a slotted spoon and freeze so that they can temper, 15 to 20 minutes.
Cook's Note: You can find liquid nitrogen at your local welding supply store or an industrial gas supplier. Please follow safety guidelines when using liquid nitrogen. If you do not want to use liquid nitrogen, you can simply mix the ginger pearl ingredients together and dollop 1 teaspoon onto each oyster before adding the mignonette.