Preheat oven to 350° F (180° C). Spray or lightly oil two 9-inch (1.5-L) round cake pans.
Grate carrots using a food processor. Measure out 5 cups (1.25 L). Finely chop ginger. Grate peel from orange. Set aside separately. Place flour in a medium bowl. Sprinkle with baking powder, baking soda, salt, and ginger. Stir until blended. Sprinkle in orange peel while stirring.
Place flour in a medium bowl. Sprinkle with baking powder, baking soda, salt, and ginger. Stir until blended. Sprinkle in orange peel while stirring.
Place butter in a large mixing bowl and beat with an electric mixer until creamy. Gradually beat in sugar, then beat on medium for 3 minutes. Beat in eggs, then vanilla. Don't worry if it seems curdled.
Beating on low, gradually add about a third of the flour mixture. Beat until just mixed, followed by half the milk.
Repeat additions, ending with flour. Sprinkle chopped ginger and stir to distribute. Stir in carrots.
Divide batter between pans. Spread as evenly as possible to sides of pan. To remove air pockets, bang pans on counter 5 to 6 times. Bake until centers seem set when lightly tapped, 30 to 35 minutes.
Place on a baking rack to cool. After about 15 minutes, turn cakes out of pans and cool completely on racks. TOP TIP: It's best to bake cakes a day ahead of icing and leave at room temperature overnight.
To make icing, beat butter in a bowl with an electric mixer until very creamy. Place chocolate in a small microwave bowl. Microwave on medium for 1 1/2 minutes. Stir, then microwave on medium for another minute. Stir until smooth. ALTERNATIVE: Or on stove top heat chocolate in a bowl set over a pan of simmering water. Stir, often. Gradually beat into butter. Add vanilla, almond extract, and salt. Gradually beat in icing sugar.
To assemble, place 1 cake, flat-side up, on a platter. Generously spread with icing, leaving a narrow border of cake around the edge.
Top with the second cake, flat-side down, and gently press down. Smoothly spread icing over the sides of the cake, then the top. Garnish top of cake with chopped crystallized ginger. It's best to refrigerate several hours or overnight before serving.