Make the fig spread: Bring the figs, 1 cup water, the balsamic vinegar and sugar to a simmer in a small saucepan over medium-low heat. 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.
For the sandwich build: Preheat a griddle over medium-low heat or heat a panini press. Split 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, then place one-quarter of the mortadella on top. Close the sandwich and lightly brush the outside of the sandwich with olive oil. Cook until golden and warm, about 4 minutes per side on a griddle or about 8 minutes total in a panini press. Repeat with the remaining ingredients.
Cut each sandwich into quarters to make 16 pieces.