It looks like your faucet is similar to our home kitchen one. That center "gizmo" is a cartridge type valve. Remove the sleeve over the large nut that holds the cartridge in place, then the nut. On ours the outer ring isn't threaded on so it lifts off easily and the large nut holds the cartridge tight in the unit. You can then see if there's anything wrong below the cartridge. Ours is a Moen with lifetime warranty. We called Moen and they sent us a new cartridge for free. The bad news is our faucet is still dripping so...we are guessing the cartridge is bad so will be calling Moen back!
But you can always just replace the whole faucet. Good luck.