To prepare rolls, dissolve the yeast in warm water in a large bowl; let stand 5 minutes. Lightly spoon flour into dry measuring cups; level with a knife. Add 1 1/2 cups flour to yeast mixture. Cover with a damp towel; let stand at room temperature 24 hours.
Add 1 1/4 cups flour, sugar, and salt to yeast mixture; stir well to form a stiff dough. Turn dough out onto a lightly floured surface. Knead until smooth and elastic (about 8 minutes); add enough of remaining flour, 1 tablespoon at a time, to prevent the dough from sticking to hands. Let dough rest 5 minutes. Knead in figs and pecans.
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 1/2 hours or until doubled in size. (Gently press two fingers into the dough. If indentation remains, dough has risen enough.) Punch dough down; cover and let rest 5 minutes.
Divide dough into 20 equal portions. Working with 1 portion at a time (cover remaining dough to prevent drying), shape each into a ball. Arrange balls on a baking sheet lined with parchment paper. Let rise in a warm place (85°), free from drafts, for 1 hour or until dough is doubled in size.
Preheat oven to 425°.
Bake rolls at 425° for 10 minutes or until lightly browned. Cool rolls on a wire rack.
To prepare butter, place the butter in a small bowl; beat with a mixer at high speed until light and fluffy. Slowly add the maple syrup, beating until blended. Serve butter with rolls.