No, that trick doesn't work.
Schengen has a 90-of-180 rule, which says you can only be in Schengen 90 days of the last 180 days. If you stay 90 days in Schengen, you must then leave to avoid an overstay.
However, Poland has a special rule allowing 90 additional days for Americans. So you can use up your Schengen 90 days, then, go another 90 days in Poland.
You can spend an intervening day in Ukraine if you really want to, but it's not necessary.
While inside Schengen (including Poland), you accrue time on both clocks.
Suppose your start in Poland, spend 90 days, then go to Germany. Nuh-uh. You already accrued your 90 days inside Schengen, while you were in Poland, and now you must leave the Schengen area for 90 days prior to visiting Germany again. You could stay 90 more in Poland because of the exception.
The 90-of-180 rule means if you spent 90 of the last 180 days inside Schengen, you are not admissible.
Suppose you enter the EU and spend 150 days in Poland. How many of the last 180 days have you spent in Schengen? 150. That makes you ineligible to enter any of the other Schengen countries (except Denmark). Now you spend a day in Ukraine. How many of the last 180 days have you spent in Schengen? Still 150.
Suppose you spend 45 days in Germany, 100 days in Ukraine, then 60 days in Poland. In the last 180 days, 100 days were in Ukraine and the rest were in Schengen, so 80 days in Schengen. You can spend 10 more days in Belgium, but then you have to leave Schengen (unless you avail yourself of the extra 90 days offered by Denmark or Poland).