Boil a large pot of salted water and cook fettuccine until al dente.
Steam broccoli florets until tender-crisp, or add to pasta water in the final 3 minutes.
Drain pasta and broccoli, reserving 1 cup of pasta water.
Heat olive oil and 1 tablespoon of butter in a large skillet over medium-high heat.
Season chicken with salt and pepper and sear until golden and cooked through, about 6-8 minutes.
Remove chicken to a plate.
In the same skillet, melt remaining butter and sauté minced garlic until fragrant.
Pour in heavy cream and simmer until slightly reduced.
Whisk in freshly grated parmesan until the sauce is smooth and thickened.
Return chicken and pasta-broccoli mixture to the skillet.
Toss everything together, using reserved pasta water if the sauce needs thinning, and serve hot.