Don't know what it's called!!!!!

You are looking to make a round tenon. You can do this several ways.

First you can use the tenon; dowel; plug cutters you can find most anywhere that sells woodworking tools. These tool are mostly made for making plugs but could be used for tenons. You may find you still have to cut off the outer part as the tool may not cut all the way to the edge of the post.

Second way is with a true tenon cutter. These tools are just for making tenons and are much better built. They also cost more than twice what a plug cutter cost.

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Third way is to use a router and a straight cutting bit. You need to make a jig to do it this way. Make a wooden box. Cut a hole in one side that your pickett will just fit in. On the other side cut a hole the size of the tenon. In your case 1". In the top cut a hole for the router bit to stick through and mount the router to the box. Adjust the bit so that when you turn the pickett it cuts all but 1" center to make your tenon. The cut tenon comes out the other side. You could put a stop on it so every tenon is the same length.

This is also a great way to make long dowels out of different kinds of wood.

The joint you're wanting to cut is a tenon, requiring a tenon cutter. Tenon goes into the (round) mortise in the rail. Not a very strong joint, but may be adequate for you. Windsor chairs that use a similar joint have a tapered tenon which then self-tightens any time somebody sits in it.

I hope you're talking about the top rail because that joint into a bottom rail will give a great path for rain to get in and greatly decrease the lifespan. If you're using a bottom rail, a better joint would be to cut an inverted V on the rail and notch the picket bottoms to fit.

Jim