Thank you everyone for your quick responses. I plan on making these this Sunday, and I am very excited!

It'll be fine with a mixer, or by hand.

You absolutely, positively don't need a machine of any kind to make the dough for these cookies. The recipe calls for butter at room temperature, and it's a shortbread dough. When I make shortbread, I often start with soft butter and I squish in all the other ingredients with my hands. Depending on the cookie, I either roll the dough into a log while it's soft, then wrap it, refrigerate it and slice and bake it; or I'll refrigerate the dough briefly and roll it into balls after the butter has firmed up, then bake the balls. This recipe is a perfect candidate for the slice-and-bake method.

A good trick for getting the butter in by hand is freeze the butter, then grate it on the large holes of a box grater.

The fig and blue cheese savories are like a shortbread. I face the same problem (no food processor). You can always rub the butter (start with cold, small chopped butter) into the flour with your fingers (I would add the blue cheese after the butter is rubbed into the flour). It should turn out just as well, but it will probably take you quite a long time and you may get sore arms - to get a nice dough. The flour and butter (and cheese) should look like bread crumbs and then eventually hold together in a smooth dough. Good luck!