The only tools you will need to create these beautiful eggs are tweezers and a small paintbrush. Be sure to work in a draft-free area when applying edible gold leaf so it doesn't blow away.
Place quail eggs in a medium saucepan and cover with water. Bring to a boil over medium-high heat, then immediately remove from heat. Cover pan and let eggs sit 2 minutes. Drain eggs and transfer to a bowl of ice water to stop the cooking. Let sit 3 minutes. Dry eggs completely before gilding.
To gild eggs: Working with 1 egg at a time, lightly and randomly brush with egg white. With tweezers, apply a small piece of gold leaf to egg white and gently brush with a small paintbrush to affix.
Stir together salt and matcha, and serve with eggs.