Prosciutto-Chicken Pasta

In a large pot of salted boiling water, cook linguine until al dente. Drain, reserving 1 cup pasta water, and return to pot. Meanwhile, in a large skillet over medium-high heat, heat oil. Add prosciutto and cook, stirring occasionally, until crispy, 4 to 5 minutes.Transfer to a paper towel-lined plate. Add chicken to skillet (with another swirl of oil, if necessary) and season generously with salt and pepper. Add oregano and stir until coated. Cook until golden and no longer pink, 6 minutes per side, then transfer to a plate. Add garlic to skillet and cook until fragrant, 1 minute, then add broccoli and 1/4 cup water and cook until tender, 3 minutes. Add milk, 1/2 cup reserved pasta water and Asiago to skillet and bring to a simmer to let thicken, 2 to 3 minutes. Add cooked linguine and toss until creamy, then add chicken and prosciutto and toss until combined (add more pasta water by the tablespoon if desired). Garnish with cheese and serve.