Heat 2 Tbsp. olive oil in a large skillet over medium-high heat. Add mushrooms and cook for 5 minutes. Remove from skillet; set aside.
Season lamb chops with 1 tsp. salt and pepper. Heat remaining 1 Tbsp. oil in skillet. Add lamb chops, in batches if necessary, and cook over medium-high heat, turning once, until medium-rare, 6 to 8 minutes. Transfer to a platter; cover with foil to keep warm.
Remove all but 1 Tbsp. fat from skillet. Add shallot and cook for 2 minutes. Pour in wine, scraping bottom of pan. Bring to a boil, lower heat and reduce liquid for 5 minutes. Add broth; cook for 10 minutes.
In a medium saucepan, bring chicken broth and 1 tsp. salt to a boil. Gradually pour in polenta, whisking. Reduce heat and cook, stirring, for 5 minutes. Stir in butter. Cover and set aside.
Add chops and mushrooms to red wine sauce and warm through. Serve with polenta.