The phrase "to balance the plate" means to evenly distribute or manage different responsibilities or tasks in order to maintain a stable and manageable situation. Full definition
Instead of trying to balance your plate on your lap, why not plan to set up a table that allows you eat while you can still see the feature bill? (sarahrichardsondesign.com)
There are no rules, you can balance the plate as you like. (yes-moreplease.com)