Wash the arugula, dry it, trim the stems off, and tear the leaves into bite-size pieces. Should be about 3 cups, loosely packed.
Wash the escarole, dry it, and tear it into bite-size pieces. Should be about 6 cups, loosely packed.
Bring port, sugar, and figs to boil in medium saucepan over high heat. Cover pan, reduce heat to low, and simmer until figs are very soft but not mushy, about 15 minutes. Reserving liquid in pan, remove figs with slotted spoon and, when cool enough to handle, quarter and set them aside.
Whisk vinegar, shallots, salt, and pepper into port; gradually whisk in oil. Return figs to dressing and reheat over medium heat, stirring occasionally, until warm but not steaming.
Toss greens and warm dressing to coat in large mixing bowl. Divide dressed greens among 6 serving plates, sprinkle with a portion of blue cheese, and serve immediately.