Break fettuccine into sixths (I broke chunks of noodles in half then each half into thirds.) Cook pasta in a big pot of salted, boiling water until just under al dente - it will continue cooking in the sauce. Drain then set aside.
Melt butter in a large skillet over medium heat then add carrot, celery, and shallot. Season with salt and pepper then saute until tender, 5-7 minutes. Sprinkle in flour then whisk and cook for 1 minute. Slowly pour in chicken broth and milk while whisking constantly to avoid lumps. Turn heat up to medium-high until bubbling then turn heat back down to medium, add parsley and more salt and pepper then cook, stirring occasionally, until thickened and bubbly, 5 minutes.
Add pasta to the sauce then stir to combine and cook for 2 more minutes. Remove from heat. Sauce will continue to thicken as it cools.