Preheat oven to 350°.
Place 18 paper muffin cup liners in muffin cups; coat liners with cooking spray. Set aside.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour and next 4 ingredients (through salt) in a large bowl, stirring with a whisk. Make a well in center of mixture. Combine sour cream and next 5 ingredients (through egg), stirring well with a whisk; add to flour mixture, stirring just until moist. Add carrot, currants, and pecans; stir just until combined.
Spoon batter into prepared muffin cups. Bake at 350° for 25 minutes or until muffins spring back when touched lightly in the center. Remove muffins from pans immediately, and place on a wire rack.