Trim the tops off the beets and drizzle with the olive oil. Season the beets with salt and pepper, wrap them in aluminum foil individually and roast until tender, about 40-60 minutes(depending on how large they are). When cool enough to handle, peel the beets.
While the beets are roasting, place the shallot, jalapeno, ginger, garlic, mint, cilantro, lime juice and 6 tablespoons of olive oil in a small bowl and stir to combine.
When the beets are done, carefully remove the foil. Let cool, and then peel the beets by slipping the skins off with your fingers. Dice the beets and add salsa mixture, stirring to combine. Taste for seasoning.