Bring a large pot of salted water to a boil. Add the pasta to the boiling water and cook according to the package directions.
Sprinkle the chicken on one side with salt and pepper. Heat the olive oil over medium-high heat in a large skillet. When the oil is hot, add the chicken, seasoned-side down, then sprinkle the other side with salt and pepper. Cook until golden brown and cooked through, 5 to 6 minutes per side. Remove from the skillet and set aside.
Drain the pasta, reserving 1 cup of the pasta water. Set aside.
With the empty chicken skillet over a medium heat, add the white wine, scraping the bottom of the skillet with a wooden spoon, until almost fully reduced, about 30 seconds. Add the butter and melt. Add the broccolini and saute until bright green, about 1 minute. Add the asparagus and saute for 1 minute more. Add the garlic and oregano and stir to combine. Add the cream and bring to a simmer, stirring. Stir in the grated Parmesan, lemon zest and juice. Season with salt and pepper. Add the reserved pasta to the sauce and toss well to coat the pasta, adding the pasta water 1/4 cup at a time, if the sauce is too thick.
Slice the chicken breasts thinly and place on top of the pasta, or serve them whole if you prefer. Garnish with the basil leaves and Parmesan shavings.