Clean the bowl down to the briar. Pick out the charred portions of the bowl. Inside the bowl a mixture of plaster, charcoal, and salt. Driers very hard. 2 parts plaster, 1 part charcoal, 1 part salt. Press this into the burnt portion. It may take a couple of applications. After dry, sand the repair flush.
For the outside, if this is a crack mix briar dust and CA glue and apply with a toothpick. This repair can be touched up with a furniture pen, or a jiffy marker.