Combine 1 1/2 tablespoons flour with salt and pepper in a bowl. Dredge beef strips in the mixture.
Melt 3 tablespoons butter in a skillet over medium heat. Add the dredged beef strips; cook and stir until browned, about 5 minutes. Add mushrooms, onion, and garlic; cook until onion is barely tender, 3 to 4 minutes. Transfer the beef and mushroom mixture to a plate.
Melt remaining butter in the same skillet; reduce heat to low. Stir in 3 tablespoons flour and tomato paste; cook until slightly browned, 1 to 2 minutes. Increase heat to medium. Add beef stock slowly, stirring constantly, until gravy is thickened and no lumps remain, 5 to 10 minutes.
Return the beef and mushroom mixture to the skillet with the gravy. Stir in sour cream and sherry; heat until bubbling, about 5 minutes.