HOW is the cook top wired? Is it some sort of NonMetalic? Is it some sort of armored cable?
Or is it conductors in (hopefully) 3/4" conduit of some sort?
The island would be adequately served by one 20 amp circuit.
I don't know if your new cooktop is going to be happy with only a 30 amp circuit, although I have never seen one that was not.
If you have conduit with individual conductors in it, you have no problems. Two hots at 10 ga for the cooktop. Don't know if it needs a neutral, but that would be 10 ga as well.
Nowadays we only allow metal conduit to serve as a ground path for a 20 amp circuit. Larger than that and you need a ground wire. And you almost certainly will be required to use a 10 ga. But you should check with your local inspector, he might be in a good mood and let you use 12 ga for the neutral and the ground.
So all you need now are a 12 ga hot and a 12 ga neutral for the island outlets. The two circuits can certainly share the ground. Change the old wires if they are not THHN. And use green for the ground and white for the neutrals. Don't confuse which neutral is which, tape them to the corresponding hots in both the panel and where they enter the box under the cabinet.
If, however, you cannot change the wire going into the island, do the simple thing:
Reassign the 10 ga (assuming there is a neutral) to a very small sub panel mounted in the cabinet below the cook top.
You can use one that has only two spaces (but is rated for four circuits), and buy a triplex breaker: a two pole 30 amp breaker in the middle and two single pole breakers on either side. You only use one of the 20 amp breakers.
This is a Murray sample, although this one is for outdoors. I'd be astonished if they did not have just the same thing for indoors without the hub on top or a door.
Such a box is very small, barely larger than a 4" sq junction box.
Continue to protect it at the main panel with a 30 amp breaker. You will need to be trying very hard to trip it.
And do not connect the neutrals to the subpanel's chassis. The neutral and the ground should only be bonded in one place in the system: the main service panel.
If that does not serve, let me know as I am pondering your original, somewhat convoluted question.