You are assuming all eat at the same speed except the special mouse. So simple. Give them all one cake each at the same time. 11 will finish together and 1 will finish either sooner or later.
If you have a problem in administering 12 cakes to 12 mice at the same time, you can do it in parts. Give 3 cakes to 3 mice and you know whether the special mouse is in this group or not. Repeat till you find your special mouse. The maximum number of cakes is still 12 but the expected value will come down to 6. Time is not optimized though and you have to do max 4 (expected value 2) iterations.