It is so simple, I can't believe it took me so long to figure it out.
1. Drive with your bike to Sheridan Road.,
3. Ride around on the other side of Sheridan. Toward Lake Michigan.
The older suburbs on Lake Michigan are great places to cycle. On the other side of Sheridan, there is no on street parking (at least in places like Lake Forrest). So, park on the other side of the street and you can bike in a fabulous setting.
These suburbs don't do something gauche like have gates. They carefully filter out the traffic and people by subtle barriers. By the time you get to Sheridan, you have made it. They use all tactics -- one way streets, 3 increasingly more efficient ways of going north/south (I94, 41, Green Bay Road) to discourage through traffic. etc.
I just started biking close to my vehicle since the weather has been shaky lately. This encouraged me to just randomly park the vehicle (too embarrassingly large to admit-but a rental) on an attractive street and just go from there. I keep as close as it seems prudent to the vehicle and just explore.
The streets next to the Lake are very well maintained. The landscaping is immaculate. Very little traffic. This is the perfect place to casually cycle.