Add the buckwheat flour, kosher salt, whole milk and eggs to a blender and pulse until smooth. Cover and chill the batter overnight.
When ready to make the crepes, remove the batter from the fridge and stir.
Heat a crêpe pan or large non-stick skillet over medium heat. Add 1 tablespoon of the butter and allow to melt, shifting the pan so the butter evenly melts and greases the pan.
Pour ¼ cup of the batter in the pan, swirling as you pour the batter so that it coats the pan evenly. Cook the crêpe until golden brown and it easily lifts from the pan, about 1 to 2 minutes. Use a spatula to cook for an additional 1 to 2 minutes then remove to a plate. Continue with the remaining batter.
Stack the crêpes on top of each other and cover with a towel to keep warm.
For assembly:
Heat a large non-stick skillet over medium heat, add olive oil and heat through. Crack eggs into pan working two at a time, and fry until the edges are golden brown, about 2 minutes. Set eggs aside and repeat with the remaining eggs.
Working one at a time, reheat the crêpes in a dry, non-stick skillet and top with ¼ cup of the shredded gruyere in the middle. Add a slice of the ham and top with one of the fried eggs and avocado. Repeat with the remaining crêpes and ingredients. Garnish with arugula.