So I went to the internet, and found that others had had a similar dilemma, and most seemed to solve it by linking to this intense pesto recipe from Dorie Greenspan. A simple idea: instead of basil, use garlic scapes!
As I sat down with my heap of scapes, I realized that while the recipe called for about ten stems, the pile I was confronting had about fifty. So I upped the amounts a bit, and hoped the garlic wouldn't overwhelm the other flavors completely.
Half a pound of walnuts, toasted.
Half a pound of parmesan cheese and a whole lot of olive oil (I stopped pouring when the texture seemed right).
We went to Claudio's, the local Italian cheese wonderland, to get the parmesan (and a few other things, since I have no self restraint when it comes to cheeses), and when the cheese monger heard what we were making, we were treated to a story about the garlic-y Italian sisters he and his friend once dated.
I put a little too much in my mouth the first time I tried the pesto - it's pretty intense right out of the food processor. But not in a bad way.
And the color is lovely.
It was delicious. But, as the fellow at Claudio's said, not a first-date kind of meal.