Tap Air Portugal has round-trip flights from Chicago (ORD) to Milan (MXP) for $382-$417 with a connection. Flights in February, April, and May.
Note: Some small OTAs on Skyscanner have this for about $368. 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. Be advised that TAP charges about $95 each-way for a regular sized checked bag for their “Discount” fare class on their International flights to/from the USA. Upgrading to a higher fare class that includes a carry-on bag (for about $60 each-way) may be cheaper than booking the discount fare with the added carry-on bag fee.