Creating a list of dates

Questions
1

I’m using the following code to create a list of dates:

DateString[#, {"MonthNameShort", ". ", "DayShort"}] & /@ 
 DayRange[{thisyear}, {thisyear, 12, 31}]

How can I change the elements of DateString so that on the first of the month, it only shows the MonthNameShort, and on subsequent days of a month it shows the DayShort?

2 
DateString[#, 
   If[#[[3]] == 1, {"MonthNameShort", ". ", "DayShort"}, 
    "DayShort"]] & /@ DayRange[{thisyear}, {thisyear, 12, 31}]