Tap Air Portugal has round-trip flights from Chicago (ORD) to Milan (MXP) for $403 with a connection. Flights January to May.
Some small OTAs on Skyscanner have this for cheaper than the airline ($385). If you plan on booking there, beware of any site that charges a fee to cancel a ticket within the 24-hour “free” period. Use unknown booking sites at your own risk.
Carry on bag with a maximum weight of 17.6 lbs is allowed for free on TAP. You can avoid seat selection fees by accepting the randomly assigned seat at the gate. Upgrading to a higher fare class that includes a carry-on bag for $65 may be cheaper than booking the “Discount” fare with the added $95 carry-on bag fee.