Without using @Ray Dehler suggestion to use a jQuery hack, I think you'd need to try creating a dependent picklist which would wind up being another hack of sorts, especially if you needed the data to still be in proper Date Format.
If you did that, you'd end up where you had a final multi-select picklist (for days) where the values would then need to be assembled with year and month using a formula to put them into date format. I don't think that would be a very efficient use of the database nor would it be particularly user friendly.
Best Answer
Use this.. instead
{!now()}
pass your custom field{!CutomObject.FieldAPIName__c}
Output