Melt 1 Tbsp each of butter and oil in a deep large nonstick frying pan over medium-high heat.
Sprinkle the chicken with salt and pepper. Add the chicken to the hot pan and cook until pale golden and just cooked through, about 4 minutes per side. Transfer the chicken to a plate to cool slightly. Coarsely shred the chicken into bite-size pieces.
Meanwhile, add 1 Tbsp each of butter and oil to the same pan. Saute mushrooms, onion, garlic, and thyme, until the onion is translucent, about 5 minutes. Transfer the mushroom mixture to the plate with the chicken.
Melt 3 more Tbsp butter in the same pan over medium-low heat. Add the flour and whisk for 2 minutes. Whisk in the milk, cream, broth, nutmeg, and a dash of salt and pepper. Increase the heat to high. Bring to a boil and simmer, uncovered, until the sauce thickens slightly, whisking often, about 10 minutes.
Bring a large pot of salted water to a boil. Add the linguine and cook until it is tender but still firm about 9 minutes. Drain.
Add the linguine to the sauce, along with the chicken and mushroom mixture. Toss until the sauce coats the pasta and the mixture is well blended. Transfer the pasta mixture to the prepared baking dish.
Stir the cheese and breadcrumbs in a small bowl to blend and sprinkle over the pasta. Dot with the remaining 3 Tbsp of butter. Bake, uncovered, until golden brown on top and the sauce bubbles, about 20 minutes.
