Stir 1 garlic clove, finely chopped, ¼ cup extra-virgin olive oil, ¼ cup (loosely packed) finely chopped parsley, and a large pinch of kosher salt in a small bowl to combine. Set garlic mixture aside at room temperature until ready to serve.
Heat ½ cup extra-virgin olive oil in a large Dutch oven or other heavy pot over medium. Working in batches if needed, add 8 oz. baguette or country-style bread, halved lengthwise, cut side down, and cook until deep golden brown underneath, 5–8 minutes, depending on thickness. Transfer bread to a plate, leaving oil behind.
Reduce heat to low. Add 1 medium onion, sliced, and 6 garlic cloves, coarsely chopped, to pot and cook, stirring often, until onion is softened and golden brown around the edges, 5–8 minutes. Add 2 Tbsp. chicken or vegetable bouillon paste, 1 Tbsp. smoked paprika, 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, ¼ tsp. crushed red pepper flakes, and 5 cups water. Stir to combine, then bring to a simmer. Tear toasted bread into large (about 3″) pieces and add to pot. Increase heat to medium-high, cover, and cook, adjusting heat as needed to maintain a simmer, until bread is swollen and very soft, 15–18 minutes. Remove from heat.
Using an immersion blender, carefully blend soup in pot until very smooth. (Alternatively, you can blend in batches in a regular blender, transferring to a medium bowl as you go. Return soup to pot before proceeding.) Return pot to medium-low heat and reheat soup, stirring and thinning with more water if needed (soup should be the consistency of heavy cream). Stir in 1 tsp. sherry vinegar or red wine vinegar; season with freshly ground pepper. Taste and season with more salt if needed. Reduce heat to low, cover, and keep warm until ready to serve.
Heat 1 Tbsp. extra-virgin olive oil in a large nonstick skillet over medium. Crack 3 large eggs into skillet and cook until whites are just set, about 4 minutes; transfer to a plate and season with salt and pepper. Repeat with remaining 1 Tbsp. extra-virgin olive oil and 3 large eggs.
Ladle soup into bowls and top each with a fried egg and some reserved garlic mixture. Season with more pepper. Finely grate Parmesan or Manchego cheese over if desired.