Cook the fettuccine according to box directions.
In a large skillet over medium heat, heat oil. Add garlic and cook until fragrant, 1 minute. Sprinkle flour over evenly, then stir and cook until mixture is lightly golden.
Very gradually add broth in while whisking, 2 tablespoons at a time, waiting for mixture to become completely smooth before adding more broth. Bring mixture to a boil, then gradually stream in milk while whisking. Bring to a simmer and cook until sauce is thickened, 2 to 3 minutes.
Remove from heat and add Parmesan and yogurt, if using. Season with salt, pepper, and a pinch of red pepper flakes.
Add pasta and toss to combine. Enjoy!