Disposal operation forces drain water into second sink

I have a double kitchen sink that the contractors installed several years back. See picture. The main sink is on right side and smaller sink is to left. They installed the disposal to the main sink because space was tight and we have pull out trash drawer in space under shallower left hand sink. Our galley kitchen is small and we were trying to get the most use from the space.

Now when I run the disposal, nasty water shoots back up in to the left hand sink. At first I thought there might be a clog in the wall after the small drain but if there is a blockage, it is short lived because once I run the disposal for less than 30 seconds, the water goes back down in small sink and all is fine with no slow water draining in either small or large sink drains. I just want to eliminate that initial burst of nasty water in the small sink. I think the problem may have to do with air pressure due the design of the connections. There are separate connections to the wall for the sinks as you can see from the attached. And the main stack and vent are located about 4ft to left of double sink.

I am wondering if it would work better to connect the small sink to the same drain as the large sink if I could get fit together in the tight space…. I appreciate suggestions.

PS there is no sewer gas smell coming from the small sink as long as I remember to run a little water to fill that P trap before I run the disposal. But if I don't run water in the Left sink P Trap, I still get the nasty water shooting, but it just comes with the lovely aroma of sewer gas.