Using a knife, trim off stems and leaves from beets and set aside. Scrub beets well to remove any dirt.
Place beets in a small or medium pot and cover with 2 inches of water. Add a large pinch of salt and bring to a boil. Reduce heat and simmer until tender and easily pierced with a knife, about 25 minutes. Drain beets and rinse with cold water. When cool enough to handle, remove skins and thinly slice.
Meanwhile, wash leaves and stems well and thoroughly dry. Remove stems from leaves and finely dice. Tear or roughly chop greens into 1-to-2-inch pieces, you should have about 2 packed cups.
In a medium skillet, heat 1 tablespoon olive oil over medium-high and add the shallot and beet stems. Cook, stirring frequently, until the shallot and stems are softened, about 3 minutes. Add the beet greens and ¼ cup water. Cook, tossing, until the greens are wilted and the water evaporates, about 3 minutes. Season with salt and pepper and remove from heat.
In a medium bowl, whisk the tahini with the vinegar and 3 tablespoons of water. Season with salt and pepper.
Spread toasts with the tahini. Top with greens and beets. Drizzle with a little extra olive oil, then garnish with flaky sea salt and cilantro leaves. Cut each tartine into four pieces.