I smell a mildew smell the first breathe I take after I sneeze. Why is that?

Fungal sinusitis. That's an unusual symptom. You might have fungal sinusitis... Not a bid deal... Very common in south east us.. But very common. It may also be nothing... Best to be seen by an ENT and have a ct scan of sinuses based on exam.
Allergy vs infection. If you are allergic to mold (a common allergy), when you encounter this in the air, it may make you sneeze. You may then notice the smell. Sometimes a sinus infection can give you a foul smell. If you have symptoms of facial pain/pressure, discolored drainage, see your doctor for treatment.