Step 1

Cook egg noodles in a large pot according to package directions. Drain and return to pot; toss with 1 tablespoon butter. Partially cover with lid to keep warm.

Step 2

Meanwhile, heat 3 tablespoons oil in a large, high-sided skillet over medium-high. Add mushrooms in an even layer; cook, undisturbed, until starting to crisp and brown, 3 to 4 minutes. Toss and continue to cook until browned in spots, about 4 minutes. Add remaining 1 tablespoon oil and toss. Cook, undisturbed, until mushrooms are browned and tender, 3 to 5 minutes.

Step 3

Add ¾ teaspoon salt to mushrooms and stir to combine. Add broth and half-and-half; bring to a simmer. Reduce heat to medium. Simmer until mixture starts to thicken and reduce slightly, 8 to 10 minutes. Remove from heat. Add vinegar and remaining 2 tablespoons butter. Fold in chicken, 2 tablespoons tarragon, and remaining 1 teaspoon salt. Top noodles with mushroom mixture and remaining 1 tablespoon tarragon.