In a medium microwave-safe bowl, combine the shrimp shells, clam juice, chicken broth, bay, thyme and peppercorns. Microwave on high until the shrimp tails have turned pink and the mixture is very hot, 4 to 5 minutes. Pour through a fine mesh strainer set over another medium bowl; discard the solids in the strainer.

Season the shrimp with salt and pepper. In a large pot over medium-high, heat 1 tablespoon of oil until barely smoking. Add half the shrimp and cook without stirring until well browned, 2 to 3 minutes. Transfer to a large plate. Repeat with 1 tablespoon of the remaining oil and the remaining shrimp.

Return the pot to medium-high and add 1 tablespoon of the remaining oil. Add the tomatoes, onion, carrot and ¼ teaspoon salt, then cook, stirring occasionally, until the tomatoes are spotty brown and the onion has softened, 3 to 5 minutes.

Add the garlic and fregola then cook, stirring, until the garlic is fragrant, about 30 seconds. Stir in 2 cups of the shrimp broth, then bring to a simmer. Reduce to medium and cook, stirring occasionally, until most of the liquid is absorbed, 8 to 10 minutes. Stir in 2 cups of the remaining broth, return to a simmer and cook, stirring occasionally, until most of the liquid is absorbed, 8 to 10 minutes. Stir in the remaining 1 cup broth and cook, stirring constantly, until the fregola is tender and the mixture is creamy but not soupy, another 6 to 8 minutes.

Off the heat, stir in the shrimp and accumulated juices, the remaining 1 tablespoon oil, the lemon juice and parsley. Cover and let stand until the shrimp are opaque throughout, 5 to 7 minutes. Taste and season with salt and pepper.