Chop off the beet greens and set them aside. Peel the beets and either grate them on a box grater or in a food processor. You should have about 2-3 cups of shredded beets. Wash the beet greens and chard. Shake to remove excess water, but no need to dry. Remove the tough center ribs and give them a rough chop. Stack the leaves on top of each other and slice them crosswise into thick ribbons. Heat the olive oil in a large dutch oven or pot on medium heat. Add the stems and cook until softened, about 5 minutes. Add the leaves of the chard and season with salt and pepper. Cook for 6-8 minutes, stirring occasionally, until the leaves have wilted and taste very tender. If the leaves still taste bitter, season with additional salt. Transfer the chard to a large bowl and allow to cool to room temperature. To make the dressing, whisk together the balsamic vinegar and brown sugar. Season with salt and pepper, then stream in the olive oil while whisking vigorously. Combine the greens and the shredded beets, and toss with the vinaigrette (depending on how much chard you have and how saturated you want your greens, you may have some vinaigrette left over). Serve, topping each plate with goat cheese. This salad will keep refrigerated for up to one week. Wait to top with goat cheese until serving.