Mound anchovies, garlic, and ¾ teaspoon salt on a cutting board. Using chef’s knife, mince the anchovies, garlic and salt together until well combined. Continue to work mixture, mashing and scraping with the side of the knife, until a smooth paste forms. (Alternatively, you can use a mortar and pestle to do this or use grated garlic and anchovy paste instead.)
Whisk tahini, lemon juice, and Dijon in a large bowl. Place the bowl on a damp kitchen towel or paper towel to hold in place.
Start slowly dripping oil into tahini mixture while whisking constantly and vigorously. Continue, slowly adding oil and whisking vigorously, until mixture looks slightly thickened and glossy. Continue to gradually add oil in a slow, steady stream until all oil has been added and mixture looks like mayonnaise.
Add a dash of water and whisk, adjusting with more water as needed, until dressing is the consistency of heavy cream. Add anchovy mixture and Parmesan and whisk until smooth. Season with salt and pepper to taste.
Transfer cooked Brussels sprouts to bowl with dressing and toss to coat.