Whisk egg yolk, mustard, and 1 teaspoon lemon juice in a small bowl. Whisking constantly, gradually drizzle in grapeseed oil, then olive oil, drop by drop at first, until mayonnaise is thickened and smooth. Whisk in anchovies and remaining lemon juice; season with salt and pepper. Cover and chill.
Do ahead: Mayonnaise can be made 1 day ahead. Keep chilled.
