Preheat your oven to 300°F. Toast pistachios and garlic cloves on a small sheet pan for 5 minutes or until fragrant. Remove the pan from the oven and allow to cool for one minute.
Add the seeds and garlic to the carafe of a blender or food processor, along with the basil and lemon juice. Blend everything together until the basil leaves start to break up a bit.
Once the basil starts to break up, remove the plug from your blender's lid and start drizzling in the olive oil. Continue to blend the sauce until you have a smooth paste.
Finally add the asiago cheese and pulse a few times to incorporate it into the sauce. Season to taste with a hearty pinch of salt and taste. Add more salt if desired.
Notes
To Freeze: fill a plastic freezer bag with the sauce, squeeze any excess air from the bag, then seal it. The pesto will keep, frozen, for up to two months.Use your pesto any number of ways: as a marinade for poultry or fish, a dipping sauce, to cover pizzas, or in any pasta dish you wish.