They say it's between 60 days or more but again there is no definite time line. You should check with your asylum office where you had filed your claim.
You can file for a travel permit that should take atleast 2-3 months to get it and cost is 165.00USD valid for 1 year.
I don't think you should use your H1B if you are travelling outside the US as it will make it confusing for the officals when you come back.
Have you applied for your EAD if not you should do it as they are taking very long to issue those specially the ones who are asylees.
I was granted Asylum in Nov 2004 and applied for EAD and still waiting.