How to Make It
In a medium bowl, cover the tomatoes with water and let stand until plumped, about 30 minutes.
Meanwhile, on a work surface, lightly pound each lamb chop to a 1/2-inch thickness. Season the lamb with salt and pepper. Spread the bread crumbs in a shallow bowl and season them lightly with salt and pepper. In another shallow bowl, beat the eggs with the parsley, garlic and a pinch each of salt and pepper.
Transfer the tomatoes and 1/4 cup of the soaking liquid to a food processor and puree until a paste forms. With the machine on, drizzle in the olive oil until incorporated. Add the grated cheese and pulse. Season the sun-dried-tomato pesto with salt and pepper and transfer to a bowl.
Dredge a lamb chop in the bread crumbs, tapping off the excess. Dip the lamb in the beaten egg, letting any excess drip back into the plate. Then dredge it again in the bread crumbs. Transfer to a baking sheet. Repeat with the remaining lamb chops.
In a large, deep skillet, heat 2 1/2 inches of vegetable oil to 350°. Set a rack on a baking sheet. Add 2 or 3 chops to the oil and fry over moderately high heat, turning, until the crust is golden and the meat is medium, 3 minutes; using tongs, submerge the bones of each chop in the hot oil for at least 15 seconds. Transfer the lamb chops to the rack and fry the remaining chops. Serve with the sun-dried-tomato pesto.
The sun-dried-tomato pesto can be refrigerated for up to 3 days. Return to room temperature before serving.