Bring the figs, 1 cup water, balsamic and sugar to a simmer in a small pot. Simmer until the figs rehydrate and the liquid reduces and thickens, about 15 minutes. Let cool. Once cool, place in a food processor and puree until smooth, 20 to 30 seconds.
Preheat a griddle over medium-low heat or heat a panini press.
Slice 1 focaccia square in half. Place a thin layer of the fig spread on the inside of both halves. Place a couple slices of cheese on the bottom, and then place 4 ounces mortadella on top.
Close and lightly brush the bread with olive oil. Griddle the sandwich until golden and warm, about 4 minutes a side. Repeat with the remaining ingredients.
Grab daintily with one hand and bite like a maniac!