1. Dissolve granulated sugar and yeast in 2/3 cup warm water in a small bowl; let mixture stand 5 minutes. Stir in 3 tablespoons melted butter.
2. Weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine 6.75 ounces (about 1 1/2 cups) flour, salt, and nutmeg in a large bowl, stirring with a whisk. Add yeast mixture to flour mixture; stir until a soft dough forms. Turn dough out onto a lightly floured surface. Knead until smooth and elastic (about 4 minutes); add enough of the remaining flour, 1 tablespoon at a time, to prevent the dough from sticking to hands. Place dough in a large bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in size.
3. Combine 1/2 cup brown sugar, syrup, and milk in a small saucepan; bring mixture to a boil. Remove pan from heat; stir in figs. Sprinkle walnuts evenly into a 13 x 9–inch baking pan coated with cooking spray; spoon fig mixture evenly over nuts in bottom of pan.
4. Combine remaining 1/4 cup brown sugar and cinnamon in a small bowl; set aside.
5. Preheat oven to 375°.
6. Punch dough down; let rest 5 minutes. Roll dough into a 12 x 10–inch rectangle on a lightly floured surface. Brush remaining 2 tablespoons butter over dough, leaving a 1-inch border. Sprinkle dough with cinnamon mixture. Roll up rectangle tightly, starting with long edge, pressing firmly to eliminate air pockets; pinch seam to seal (do not seal ends). Cut into 12 (1-inch-wide) slices. Place slices, cut side up, into pan. (Slices do not fill pan but will once dough rises.) Cover pan with a damp towel; let rise in a warm place (85°), free from drafts, 15 minutes or until doubled in size.
7. Bake at 375° for 15 minutes or until buns are lightly browned. Cool buns in pan on a wire rack for 5 minutes. Place a serving platter upside down on top of pan; invert buns onto platter. Serve warm.