Mix gelatin and 3 Tbsp. cold water in a cup; set aside. Stir together 3/4 cup coffee and 3 Tbsp. sugar in another cup; set aside.
Combine milk, remaining 1 cup sugar, salt and cinnamon in a heavy saucepan. Bring to a simmer over medium heat, whisking occasionally.
Whisk together egg and yolks in a heat-proof bowl; whisk in half of hot milk mixture. Pour whole mixture back in saucepan. Cook, whisking, until mixture reaches 170°F on an instant-read thermometer. Remove from heat, stir in gelatin, remaining 1/2 cup coffee and vanilla extract. Let cool, whisking occasionally.
Transfer cooled mousse to a bowl put in an ice-water bath. Whisk until cold, about 5 minutes. In another bowl, beat 1 1/2 cups cream until stiff. Fold into custard to finish mousse.
Line a 9-by-5-inch loaf pan with plastic wrap. Arrange 6 ladyfingers in bottom. Drizzle with 2 Tbsp. sweetened coffee, from Step 1. Layer 2 cups mousse and 12 cookies. Drizzle with 6 Tbsp. coffee. Add remaining mousse and 12 cookies; drizzle with remaining coffee. Wrap in plastic. Chill for at least 3 hours.
To serve, unmold onto a platter. Whip 1 cup cream and confectioners' sugar until stiff and spread on cake. Dust with cinnamon, and slice.