How do I book restricted income?

This is a very vague question be more explicit

That sounds like an unusual thing to do but you should be able to accomplish it by creating a new account in your chart of accounts called "restricted income" and then use the account when entering your transactions. You can link your income items to the restricted income account or make adjusting entries at the close of each accounting period.

If I debit cash and credit Income (my Income St) and then want to move it to Balance Sheet to restricted income (for later use), what account do I then Debit?
Or if I originally debit cash and credit Balance sheet income restricted income account, how does it ever hit my Income Statement?

Your use of the term "restricted income" is the thing that is causing some confusion. Cash is a balance sheet account; it's an asset. The account you affect when you debit cash while crediting income is generally your bank account. So it's already there on your balance sheet. If you then want to move, or set aside, some cash for later use, you would be debiting one asset account and crediting another.

Or put another way, income accounts aren't balance sheet accounts. So if you debit cash and credit an income account (regardless of what it's called), it IS hitting your income statement, not your balance sheet.

Either income is earned or it is not, if this is something that has a future maturity it doesn't belong in income, perhaps it is an intangiable