Put the basil and oils in a liquid measuring cup and blend with an immersion blender until coarsely blended. Cover and steep this for 24 hours.
After 24 hours strain the oil through a fine mesh sieve lined with cheesecloth. Set the oil aside and discard the basil.
In a wide-mouthed jar blend the egg, egg yolk, 1 t-spoon of lemon juice, garlic, and salt on medium speed for about 1 minute to aerate the egg and bit and combine everything. Now, with the blender still running, slowly stream the basily oil (plunging the immersion blender up and down a bit). Start by doing it drop by drop, slowly increasing it up to a thin, steady, slo-mo stream. Once it's smooth, thick, and perfectly emulsified check it for seasoning and add more salt or lemon juice as needed.
Store in the fridge for up to 2 weeks.
If you don't have an immersion blender you can just roughly chop the basil and steep in the oil. You can use either a handmixer or standup mixer with the whisk attachment to make the mayo. A traditional blender or even food processor will also work in a pinch but I prefer the former.