Preheat the oven to 350°. In a small baking dish, toss the beets with the olive oil, thyme, garlic and 1/4 cup of the vinegar and season with salt and black pepper. Cover tightly with foil and roast for about 1 1/2 hours, until fork-tender. Let cool slightly.
Meanwhile, in a small saucepan, combine the cherries with the fennel seeds, sugar, 3/4 cup of water and the remaining 1/4 cup of vinegar and bring to a simmer. Remove from the heat and let stand for 30 minutes.
In a large saucepan, heat 2 inches of canola oil over moderately high heat until shimmering. Add the sliced shallots and cook over moderate heat, stirring, until they are golden brown, about 10 minutes. Using a slotted spoon, transfer the fried shallots to paper towels to drain.
Peel the beets and slice them 1/4 inch thick. Strain the beet cooking liquid into a large bowl. Spoon 2 tablespoons of the cherry pickling liquid into the bowl along with the lemon juice and whisk to combine. Season the dressing with salt and black pepper. Strain the cherries and discard the remaining pickling liquid.
Arrange the beets on a platter. Toss the mizuna and cherries in the bowl with the dressing. Mound the greens over the beets, garnish with the fried shallots and serve.
The beets can be steamed one day in advance and refrigerated.