Place a rack in center of oven; preheat to 350°F. Coat a 9x9" baking pan with nonstick spray. Line pan with 2 overlapping layers of parchment, leaving a 2" overhang on all sides. Coat parchment with nonstick spray.
Whisk sugar, cocoa powder, and salt in a large bowl. Melt butter in a medium heatproof bowl in the microwave or in a medium pot over medium heat. Immediately add hot butter to sugar mixture and stir with a wooden spoon or rubber spatula until smooth. Add eggs one at a time, stirring vigorously with spoon or spatula after each addition, until batter is thick, shiny, and smooth. Stir in molasses and vanilla. Using a sifter or fine-mesh sieve, sift in flour, cinnamon, ginger, baking powder, and cloves. Stir until well incorporated, then beat vigorously for 40 strokes.
Transfer batter to prepared pan; smooth top. Bake brownie until slightly puffed and top is firm to the touch, edges are set, center is moist but not uncooked, and a tester or paring knife inserted into the center comes out with moist crumbs (not runny batter) attached, 40–45 minutes.
Transfer to a wire rack and let brownie cool completely in pan. Using parchment overhang, lift brownie out of pan and transfer to a cutting board. Cut into 16 squares.