Meanwhile, bring a 2-qt. pot of salted water to a boil over high heat. Add the asparagus tips and cook until just tender, 2 to 3 min. Drain in a colander, shower with cold water to stop the cooking, and drain again. Set aside.
In another 3-qt. (or larger) saucepan, melt the remaining 2 Tbs. butter over medium-low heat. Add the white and light-green sliced leek and the remaining celery and season with a generous pinch of salt. Cook, stirring occasionally, until the leeks look soft but not browned, 3 to 4 min. Add the garlic and cook for 1 minute more. Add the asparagus spears and the potato. Set a wire mesh strainer over the pot and pour in the stock from the other pot; discarding the solids. Stir well and bring to a boil over high heat. Reduce the heat to medium-low, cover, and cook at a lively simmer until the potatoes and asparagus are very tender, about 20 min. Turn off the heat and let cool slightly.
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