Heat oil in a large saucepan over medium-high heat. Add caraway and fennel seeds and red pepper flake and cook, stirring, until a few seeds pop or they smell very fragrant.
Add onion and garlic, sprinkle with salt and pepper, and cook, stirring often, until onion is tender, about 5 minutes.
Add stock and beets, bring to a simmer, reduce heat, cover, and simmer for 20-30 minutes.
Transfer to a blender and blend smooth. Add balsamic vinegar and blend to combine.
Taste, and season to taste with salt, pepper, and more vinegar, if needed.
Strain through a fine mesh strainer into a clean saucepan and warm as necessary.
Serve, placing a round of goat cheese in the center and sprinkling with toasted walnuts.