You could have an extra deep counter on the old part of the wall. Either pull the cabinets out from the wall and use a ledger board to support the back edge of the counter, or get new extra depth cabinets if possible. Any uppers, including the hood, would possibly need to be pulled out as well. [ETA: To make them match uppers in the new part of the kitchen, you'd want them pulled forward.] (Not a designer or pro, so hopefully someone will chime in.) One consideration is the cooktop or range, and how you will handle the extra space behind it. There's a picture of a slide in range in a deep counter run in the second thread linked below.

Without a measured floor plan, it's tough to say if or how those pulled out cabinets will affect the rest of the kitchen.

Here are some old threads on the topic:

How to pull kitchen cabinets forward

Extra Deep Countertops