Make a buttery sauce less oily
You want emulsification. 

Emulsification is the breaking up a fat and dispersing it into a liquid (or vice versa, dispersing a liquid in a fat). A classic example of an emulsification, also known as an emulsion, is mayonnaise.

There are at least two good ways you can emulsify your brown butter, soy sauce, and lemon juice. One way is called shearing, which is just what it sounds like. You're basically cutting off slices of droplets, making them smaller and smaller. The smaller the microdroplets, the more stable the emulsion. You can accomplish a short-lived but great textured emulsion with just energetic whisking of your liquid and fat ingredients. You can make the micro-droplets even smaller and your emulsion more stable by using a blender or immersion blender.

A second thing you can do to bind your liquid and fat ingredients into a much firmer and more stable emulsion is to add an emulsifier, an egg yolk being a very solid candidate for the job. Energetically whisk in about half of an egg yolk while slowly bringing the cool sauce to just above perfect serving temperature. If you like where the sauce is going, and would like it to be even more like that, temper, then add the other half of the yolk. Mustard is pretty effective too, for a different flavor.