Heat butter and olive oil in a large deep skillet or pot over medium heat. Add chicken, season lightly with salt and pepper, and cook for 4–5 minutes until lightly golden and cooked through. Remove chicken and set aside
In the same pan, add garlic and cook for 30 seconds, just until fragrant.
Add uncooked linguine and pour in chicken broth. Bring to a gentle simmer.
Reduce heat to medium-low and cook, stirring occasionally, for 10–12 minutes, until the pasta is nearly al dente and most of the liquid is absorbed.
Stir in heavy cream and Parmesan cheese until smooth and creamy.
Add asparagus and return the cooked chicken to the pan. Continue cooking for 4–5 minutes, until asparagus is tender and the sauce thickens.
Taste and adjust seasoning. Add reserved pasta water or broth if needed to loosen the sauce.
Remove from heat and garnish with fresh herbs before serving.