Preheat the oven to 350°F (175°C). Lightly coat bottom and sides of a 9-inch round cake pan with baking spray. Line bottom with parchment paper and lightly coat paper with baking spray; set aside.

Beat together eggs, brown sugar, granulated sugar, baking powder, ginger, salt, cinnamon, and baking soda with an electric mixer on high speed until thick, pale, and increased in volume, about 3 minutes. Gradually add melted butter with mixer on medium speed until combined.

Fold in flour until just combined. Gently fold in grated carrots and toasted walnuts until evenly distributed without overmixing.

Transfer batter to prepared pan and spread into an even layer with a small offset spatula.

Bake in the preheated oven until a wooden pick inserted in the center comes out clean, about 45 to 50 minutes. Let cake cool in the pan on a wire rack for 15 minutes. Remove and discard parchment; cool completely on wire rack for 90 minutes.

Prepare the glaze: Cook sugar, carrot juice, cream, and salt over medium-high heat, gently swirling the pan occasionally, until thickened and coats the back of a spoon, about 12 to 15 minutes. Stir in butter until combined; let cool, undisturbed, for 15 minutes.

Place the cake on a plate. Stir the glaze well and pour over the center of the cooled cake, spreading to the edges with a small offset spatula.