Heat 1 tablespoon butter and the oil in a large skillet over moderate heat. Season both sides of the chicken breasts with ½ teaspoon salt and ¼ teaspoon pepper and add to the pan. Cook until golden, about 5 minutes. Flip the chicken and add the broth, dill, and ¼ teaspoon salt. Bring to a simmer and cook partially covered for 4-5 minutes or until the chicken is cooked through. Transfer the chicken to a plate and cover to keep warm. Set the skillet aside with the broth.

Meanwhile, in a large pot of boiling, salted water, cook the orzo for 8 minutes. Add the asparagus and continue cooking until the orzo is tender and the asparagus is bright green, about 4 minutes longer. Drain, return to the pot, season with ½ teaspoon kosher salt and ¼ teaspoon pepper, and stir in the butter until melted. Cover to keep warm and set aside.

In a medium bowl, whisk the eggs with the lemon juice and ⅛ teaspoon pepper until frothy. Bring the reserved chicken broth mixture back to a simmer, then set the heat to the lowest possible setting and whisk in the eggs in a thin stream. Cook over super low heat, whisking continuously for about 3 minutes and the sauce thickens. Don’t let the sauce come to a simmer or it could curdle. Taste for seasoning and adjust salt and pepper as needed.

Slice the chicken breasts and stir any chicken drippings into the sauce. Divide the orzo and asparagus between plates, top with the chicken, and ladle with the sauce. Sprinkle with lemon zest and fresh dill if desired.