Cut top quarter off garlic heads and wrap garlic in foil. Cut squash in half, lengthwise, and discard pulp and seeds. Place squash, cut-side down, on baking sheet lined with parchment paper. Cut onion into quarters and place beside squash on baking sheet.
Roast garlic and vegetables in a preheated 350°F (180°C) oven for 45 to 60 minutes, or until very tender.
Feed garlic into food processor. Scrape out squash flesh and add it, along with onions, to processor. Purée, adding a bit of stock if necessary.
Transfer vegetable purée to a saucepan and add stock. Bring to a boil and simmer gently for 10 minutes. Season with salt and pepper, to taste.
For beet splash, peel beet(s), cut into chunks and purée in processor. Add yogourt and purée until smooth. Add milk, yogourt or water if mixture is too thick.
Serve soup in wide bowls, with some beet purée “splashed” on the top.