Place the beets cut-side down in a single layer in an 11-inch straight-sided sauté pan or Dutch oven. Add the water and salt, then bring to a simmer over high heat. Reduce the heat to low, cover, and simmer until a paring knife slides into the beets with no resistance, about 45 to 50 minutes.

Move the beets to a cutting board. Increase the heat to medium-high and boil down the cooking liquid, stirring occasionally, until the pan is nearly dry, about 5 to 6 minutes. Add the vinegar and sugar, return to a boil, and stir constantly with a heat-resistant spatula until the glaze is thick enough that the spatula leaves a wide trail when dragged across the pan, about 1 to 2 minutes. Remove the pan from the heat.

Rub the skins off the beets using a paper towel once they are cool enough to handle, then cut them into 1/2-inch wedges. Add the beets, shallot, lemon zest, remaining salt, and pepper to the glaze and toss to coat thoroughly. Transfer to a platter, sprinkle with almonds, mint, and thyme, and serve warm.