More cat skinning! Flat tangs are a challenge, especially if they are fairly wide, but there are a number of ways to deal with them.
Option one would be to carefully saw the blank in half lengthwise. Then, trace the tang on one of the halves, and use a bench chisel to carve a recess for the tang. Then, glue the two halves handle back together leaving the recess as a rectangular slot - be careful to not let any glue get into the slot at this point. Turn the handle to the desired shape and glue the tang into that rectangular slot. If you are of the 'belts and suspenders' type, you could also put some pins through the reglued handle that pass through those notches.
Another option be to cut a kerf in the handle blank that is as wide as the thickness of the tang, but only goes deep enough to receive the tang. After turning the handle, glue the tang into the kerf. The challenge with this approach is that, depending on the design of the turning, portions of that kerf might be left as gaps. You could glue some thin stock into those gaps, or you could mix up a blend of sawdust (from the turning) and epoxy glue that you use as filler. In either case, you would have some hand shaping to do (sandpaper, perhaps a Dremel) to smooth the infill before finishing the handle. Alternatively, again depending on the design of the turning, the edge of the tang could be left exposed. The infill option is probably preferable because the infill would prevent the tang from trying to rotate if the tool is later used like a knife to cut cheese.
But in my experience, the easiest option is to make the flat tang into a round tang by padding it out with wood. My approach would b e to turn a scrap of wood into a dowel with a diameter equal to the width of the tang. Cut that dowel in half lengthwise, sand the cut faces to remove enough material that when the dowel is reassembled around the tang, the original diameter is restored. Then, use epoxy to glue the halves of the towel to the tang. Put it aside for the glue to cure. Then, drill a round hole in the blank with a diameter about 1/64" larger than the dowel. Make that hole as deep as you want to inset the tang. It doesn't especially matter whether you drill the hole in the blank before turning the handle, or drill the hole on the lathe as part of turning the handle. After the glue on padded tang has cured, sand any squeezeout, apply glue in the hole in the handle, and insert the round tang.
A potential issue with either the first or third options is that when you glue the tang into the handle, you need to put glue into the hole first, and then press the tang into place. If the hole is really snug (as it ideally should be), inserting the tang traps air in the hole that will attempt to push the tang back out of the hole once you release the tang. My approach to that is to apply some painter's tape on the outside of the handle, and then drill a very small hole through the tape and the side of the handle into the bottom of the hole that is to receive the tang. I use the smallest drill bit I own (#65) for that breather hole. Then, when the tang is pressed into it's hole, the breather provides a path for compressed air to escape. Excess glue in the tang hole will also be forced into the breather to seal that hole. If there is excess, the squeezeout is on the painter's tape where it can easily be peeled away after the glue cures. The remnant of the breather hole is so small that it's invisible.
