In the realm of classic French cuisine, any preparation bearing the designation grenobloise—literally, "of Grenoble", a city in southeastern France—is served with a sauce of browned butter, capers, parsley, and pieces of lemon.
2 whole skinless sole filets (about 4 oz. each),
halved lengthwise down center line
Kosher salt and freshly ground black pepper
1⁄4 cup milk
1 lemon, peeled
1⁄2 cup flour
2 tbsp. clarified butter
2 tbsp. unsalted butter (like L'Ancêtre Bio Organic)
2 tsp. capers, drained
2 tsp. coarsely chopped flat-leaf parsley
1. Season sole filets with salt and pepper; put them into a shallow dish. Cover with milk; set aside.
2. Using a knife, cut white pith away from lemon; remove segments by slicing between membranes. Cut half the segments into 1⁄2" pieces. (Reserve the other half for another use.) Put flour on a plate; season with salt and pepper; set aside.
3. Heat clarified butter in a large skillet over medium-high heat. Remove filets from milk; fold the thin, tapered ends under to create an even thickness. Dredge both sides in flour, shake off excess, and add to skillet. Cook, turning once, until golden brown, about 2 minutes per side. Transfer to plates; cover with foil to keep warm.
4. Add whole butter to skillet; cook, stirring, until it turns a deep brown and smells nutty, about 3 minutes. Remove from heat and stir in lemon pieces, capers, and parsley; swirl skillet to combine. Spoon sauce over sole. Serve immediately.