If browser windows, name your browser. There are a few tricks such as, if a second browser window does not open full screen, click maximize or drag to maximize and then click the x to close it. It should open maximized the next time. No guarantee that will work. Of course shortcut icons can be set to open as you wish. The issue can be such with tabbed browsing and you'd need to tell the browser how to open the next windows and remember your settings.
If I click on the maximize window, I will get the full screen that I want but the half screen will reappear in later use (during the same online session).