On low heat, melt butter in a medium saucepan. Add water and stir quickly and constantly until the butter has completely melted and the mixture has emulsified. The mixture should be cloudy. Use patience as too much heat will cause it to separate.
Remove from heat and immediately add lime juice, soy sauce, quail egg yolk and sea urchin. Use an immersion blender to thoroughly blend. You can use a standing blender with caution, covering the lid down with a dish towel so the hot mixture does not explode. Set aside.
Cook pasta, al dente. Add half of the sea urchin mixture to the pasta and toss to evenly coat.
Plate the pasta and spoon a little more sauce over the individual plates.
Garnish the plate with a dusting of Espelette pepper, caviar and optional raw quail egg yolk.