Adjust oven rack to middle position and heat oven to 350 degrees. Line rimmed baking sheet with parchment paper. Microwave figs and rum in bowl, covered, until bubbling, about 1 minute. Let sit until figs have plumped, about 5 minutes. Drain figs, reserving rum to stir into chocolate, if using.
Combine grits, flour, baking powder, and salt in bowl. With electric mixer on medium-high speed, beat butter, sugar, and zest until light and fluffy, about 2 minutes. Add eggs and mix until incorporated. Reduce speed to low and add flour mixture, nuts, and plumped figs; mix until just combined. Form dough into two 7- by 3-inch logs and transfer to prepared baking sheet. Bake until light golden and just beginning to crack on top, about 35 minutes. Let cool on sheet 10 minutes.
Using serrated knife, cut logs crosswise into 1/2-inch slices. Place slices 1/2-inch apart on baking sheet and bake until golden, about 10 minutes, flipping each slice halfway through baking. Transfer to wire rack to cool completely. (Cookies can be stored in airtight container for 1 week.)