In a medium saucepan over medium-high, toast the caraway seeds, stirring often, until fragrant, 1 to 2 minutes. Add 2 quarts water and bring to a simmer, then reduce to low, cover partially and simmer gently for 30 minutes.

In a large saucepan over medium, heat the oil until shimmering. Add the onion and ½ teaspoon salt; cook, stirring, until the onion is translucent, 6 to 8 minutes. Add the garlic and cook, stirring, until fragrant, about 1 minute. Stir in the flour and paprika; cook, stirring often, until a film forms on the bottom of the pan, 2 to 4 minutes.

Set a fine-mesh strainer across the saucepan with the onion. Ladle or pour half of the caraway broth into the strainer. Remove the strainer, then whisk the broth-onion mixture to smooth any lumps, scraping along the bottom of the pan. Repeat with the remaining broth; discard the seeds in the strainer.

Add ¾ teaspoon salt to the soup, then bring to a simmer, whisking occasionally. Reduce to medium-low and simmer gently, stirring occasionally, until lightly thickened, about 15 minutes. Taste and season with salt and pepper. Ladle into bowls and finish each portion with a spoonful of the pesto.

Parsley and Walnut Pesto: In a food processor, combine ½ cup walnuts, ¼ cup finely grated Parmesan cheese and 1 medium garlic clove (smashed and peeled). Process until finely ground and beginning to clump, 30 to 45 seconds. Add 2 cups lightly packed fresh flat-leaf parsley (roughly chopped), ¼ teaspoon kosher salt and ½ teaspoon ground black pepper. Process until finely chopped, 1 to 2 minutes. Transfer to a small bowl and stir in 1 teaspoon white wine vinegar.