How to Make It
Fill a stockpot with 6 quarts of water, cover, and bring to a boil.
Peel the cucumbers and halve them lengthwise. Scoop out the seeds. Slice the cucumbers crosswise 1/4 inch thick.
In a large skillet, combine the butter, cucumbers, and 1 teaspoon salt. Cook over moderately high heat, tossing frequently, until the cucumbers are tender but still slightly crisp, about 5 minutes.
Meanwhile, add 3 1/2 tablespoons salt to the boiling water and add the fettuccine. Cover partially until the water just returns to a boil, then uncover, stir the pasta, and cook until al dente. Drain, reserving 1/2 cup of the pasta water.
Add the fettuccine to the cucumbers and toss over moderate heat to coat the strands with butter. Add some of the pasta water, 1 tablespoon at a time, to moisten if necessary. Remove from the heat, add the mint and toss. Serve immediately.