Place eggs in a medium saucepan and add cold water, covering eggs by 1 inch. Set pot over high heat, uncovered, and bring to a full boil. Immediately turn off heat, cover pot, and let sit for 5 minutes.

Drain eggs, transfer to a bowl filled with ice and water, and let sit for at least 1 minute. Meanwhile, in the same pot used for eggs, combine beet juice, vinegar, shallot, and sugar, and bring to a boil. Cook until sugar has dissolved, about 5 minutes; remove from heat, and let cool.

Peel eggs and place in a quart-size glass jar. Pour beet juice mixture over eggs, and add beet slices. Close jar, and refrigerate for 24 hours.

Remove eggs from brine, and slice lengthwise. Scoop out yolks, and add them, along with chèvre, mayonnaise, and mustard, to the bowl of a stand mixer fitted with the paddle attachment. Mix on medium-low speed until combined and fluffy, 1 to 2 minutes. Add salt and pepper to taste, and mix again until combined.

Fill a piping or plastic bag with yolk mixture (if using a plastic bag, snip off a corner once filled, and use as you would a piping bag). Pipe yolk mixture into pickled egg whites. Top each filled egg with a pecan, and serve.