I swear, I've been through tier 2 basil at least 30 times and not once did I ever see him with an Avenger recipe. Which leads me to believe that particular ports are rarer than most significantly. I don't want to spend 15k on an Avenger in AH when I can just get it for 10k with Basil, and score some crowns and tokens doing JK, but I'm starting to get desperate.
So my question to is: Are particular recipes rarer than others?
And: How long did it take you until Basil stopped trolling you and gave you the Avenger recipe?
Basil, unfortunately, is pretty much completely random. Sadly, that means that every time you go there, there's an equivalent chance for every recipe, including the ones you've already seen. That means that sometimes we all have good experiences with it, but we often also have bad ones.
Personally, I found an avenger recipe the second run I ran after starting to look. I found another 3 times later and sold it for some quick crowns. But the heavy plate shield on the other hand... I waited a good 30, maybe 40 runs before seeing one.
Probability may be trolling you, but poor Basil over there is just doing his job...