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In case you haven't resolved this yet: see e.g. com.vaadin.client.debug.internal.InfoSection or LogSection, and use VDebugWindow.addSection(Section) like ApplicationConfiguration.onModuleLoad() does for the standard sections.
If you want to show server side logs, I'd recommend spending some time thinking about what you want to send from the server to the client, how and how to limit the amount of data transferred/stored on the client.