FOR THE HERB OIL AND CRISPY CAPERS: Combine oil and capers in medium bowl (capers should be mostly submerged). Microwave until capers are darkened in color and have shrunk, about 5 minutes, stirring halfway through microwaving. Using slotted spoon, transfer capers to paper towel—lined plate (they will continue to crisp as they cool); set aside. Reserve caper oil.

FOR THE SOUP: Heat oil in large saucepan over medium heat until shimmering. Add onion and celery and cook, stirring frequently, until softened but not browned, 6 to 8 minutes. Add thyme sprigs, garlic, and cayenne and cook, stirring constantly, until fragrant, about 1 minute. Add beans and their liquid and stir to combine. Reduce heat to medium-low, cover, and cook, stirring occasionally, until beans are heated through and just starting to break down, 6 to 8 minutes. Remove saucepan from heat and discard thyme sprigs.

Process bean mixture and Parmesan in blender on low speed until thick, smooth puree forms, about 2 minutes. With blender running, add 1 cup broth and butter. Increase speed to high and continue to process until butter is incorporated and mixture is pourable, about 1 minute longer.

Return soup to clean saucepan and whisk in remaining 1 cup broth. Cover and bring to simmer over medium heat, adjusting consistency with up to 1 cup hot water as needed. Off heat, stir in lemon juice. Season with salt and extra lemon juice to taste.

Stir parsley and basil into reserved caper oil. Drizzle each portion of soup with herb oil, sprinkle with capers, and serve.