Heat oil in a large, straight-sided skillet over medium-high heat. Add red onions and saute until softened, about 5 minutes. Stir in brown sugar, vinegars, lemon zest and juice, cinnamon stick, rosemary sprig, ginger, ground cinnamon, mustard, coriander, allspice, cloves, and nutmeg. Bring to a boil over medium-high heat; reduce heat to medium and simmer until onions are tender and liquid is reduced by half, about 30 minutes.

Season chutney with 1/4 teaspoon each salt and pepper. Add figs; simmer over medium-high heat until figs break down and chutney is very thick, about 20 minutes. Discard cinnamon stick and rosemary sprig.

Preheat an outdoor grill to medium heat (325 to 375 degrees F (165 to 190 C)) and lightly oil the grate. Sprinkle chicken with remaining 1/4 teaspoon each salt and pepper. Grill, turning once, until an instant-read thermometer inserted into thickest parts registers 165 degrees F (74 degrees C), 15 to 18 minutes.

Arrange chicken on a platter. Spoon 1/3 cup of the fig chutney on each breast. Top with goat cheese and garnish with basil.