In a skillet, heat the olive oil on medium heat. Add the onions and stir to coat, letting them cook until caramelized, about 20 minutes.
In a medium bowl, strain the borscht, reserving the liquid for another use (like this Pink Rimmed Gefilte Fish).
In a blender or food processor, place the beets, goat cheese, thyme and salt. Blend on high for 1-2 minutes. If the mixture is stuck, add up to 2 tablespoons of the borscht liquid, one at a time.
Assemble the crostini by smearing 2 teaspoons of the cheese mixture on each baguette slice and topping with the caramelized onions. Sprinkle the chives over the baguette slices and serve immediately at room temperature.