Stir together 3/4 cup milk and tapioca in a medium saucepan. Let stand until tapioca starts to plump, about 30 minutes.
Whisk salt, 1/4 cup sugar and remaining 2 1/2 cups milk into the soaked tapioca. Bring tapioca mixture to a simmer, stirring often over medium heat. Reduce heat to low and cook, stirring constantly until tapioca pearls are translucent, plump (about double in size) and the mixture is thick enough, 10-12 minutes. Remove from heat.
Whisk egg yolks in separate bowl. Whisk 1/2 cup hot tapioca mixture into the yolks until fully incorporated. Whisking constantly, add tempered egg yolk mixture back into remaining tapioca in saucepan.
Return saucepan to medium-low heat and cook, stirring constantly, avoiding boiling, until mixture coats the back of a spoon; 8 to 10 minutes. Remove from the heat. Stir in vanilla bean paste and let cool at room temperature for 20 minutes. The pudding will be quite loose, but thickens as it cools.
Meanwhile, place egg whites in a separate medium bowl; beat on high with an electric mixer until foamy; about 10 seconds. Gradually sprinkle in remaining 1/4 cup sugar and continue beating until soft peaks form; 3 to 4 minutes more. Gently fold into cooled tapioca mixture. Transfer to a large bowl or individual cups. Press a sheet of plastic wrap directly onto the surface. Refrigerate until chilled at least 4 hours or up to 3 days. Serve chilled.