And what new tool do I get to buy, if so? I think this is what a router is for, am I right?
Correct router is the one (although you could go traditional as do it with planes and chisels, or moulding planes). Buy router (1/4" shank palm router will be the most controllable and best suited for the job), buy appropriate bit, set up router and away you go remembering not to do a climb cut (you want to push the cutting edge so it's spinning into the cut. If you go the other way the router can run away from you).
Correct router is the one (although you could go traditional as do it with planes and chisels, or moulding planes). Buy router (1/4" shank palm router will be the most controllable and best suited for the job), buy appropriate bit, set up router and away you go remembering not to do a climb cut (you want to push the cutting edge so it's spinning into the cut. If you go the other way the router can run away from you).