Preheat the oven to 425°F (220°C). Line two large baking sheets with parchment paper.
Divide the halved tomatoes and onions between the prepared baking sheets. Keep the tomatoes in a single layer, placing them cut-side up.
Cut enough from the top of each garlic head to expose the cloves. Place each head on a piece of aluminum foil, drizzle with approximately 1 teaspoon of the olive oil and wrap tightly.
Drizzle the remaining olive oil over the tomatoes and onion. Sprinkle with 1¼ teaspoons of the salt, the pepper and oregano.
Place one wrapped garlic head on each baking sheet. Roast for 30–35 minutes, rotating the pans and switching their positions halfway through. The tomatoes should be softened and lightly caramelized around the edges.
Remove the vegetables from the oven. Carefully open the garlic packets and allow everything to cool for about 10 minutes.
Add the cottage cheese and ½ cup (125 mL) of the broth to a high-powered blender. Blend until completely smooth.
Add the roasted tomatoes, onion and any juices from the pans. Squeeze the softened garlic cloves from their skins and add them to the blender.
Remove the blender’s centre lid cap to allow steam to escape, cover the opening loosely with a folded kitchen towel and blend until smooth. Work in batches if necessary.
Pour the soup into a large pot and add the remaining 1½ cups (375 mL) broth. Heat gently over medium-low heat, stirring occasionally. Avoid bringing it to a hard boil.
Stir in 1 teaspoon of balsamic vinegar and the remaining ¼ teaspoon salt. Taste and add more balsamic vinegar, salt or broth as needed.
Ladle into bowls and garnish with fresh basil, cracked black pepper, and a drizzle of olive oil.