Palermo in March?

My first suggestion was to look at what direct flights there are from Denver to European cities, and then see which of those cities have flights to Palermo. For instance, BA to London, and then a direct flight to Palermo. However, for a direct flight, it looks like you’d have to change airports in London, which in my view would be worse than a 2nd stopover, unless you build a day or so in London to get over jetlag and be able to get to the 2nd airport at your leisure. Otherwise, you may just have to bite the bullet and accept 2 stopovers if you want to do this tour.

I would think your only options are the connections through FRA or MUC if you are looking for a 1-stop option. The only time I went to PMO, I connected in MUC but I’ll need to check if I had another connection prior to that.

I especially like the UA flight to MUC which leaves at 8:20 PM since it arrives in Europe in the early afternoon though I’m not sure if there is a good connection onwards to PMO. Run a check on google flights

I’ve had a look, and it does look like all flights from Denver to Palermo need two stops. :(
How about flying to Rome, one stop in Germany enroute, then staying one night in Rome and flying on a separate flight to Palermo the next morning.
Seems like the easiest way.
You’d need a few hours between flights when on separately issued tickets, to ensure any bags are reclaimed then rechecked.
There are a few hotels at FCO, Rome’s main airport.

Yes, Google Flights indicates any single-ticket itinerary will have two stops.

However, DIA has non-stop flights to quite a few European airports, and I bet most of them have non-stop flights to either Palermo or Catania (Sicily's busiest airport). Taking advantage of a non-stop transatlantic flight and a separate flight on a European airline means two separate tickets. It's not something I'd risk without spending the night at the connection point, and even then, I'd be nervous on the return trip because of the very high cost of a replacement one-way transatlantic ticket if I somehow managed to miss the connection.

These are the European destinations for non-seasonal flights from DIA:

Dublin
London-Heathrow
Frankfurt
Munich

A round trip to Dublin or London-Heathrow might be quite affordable. I've used the Premier Inn T4 at Heathrow several times; it's usually less than 100 GBP per night; you'd probably need two rooms. If you all happened to arrive in good shape after the overnight flight (which I never do), you could hop over to Windsor and Eton for the day if you didn't want to make a rushed trip into London.

I suspect I'd find a two-stop itinerary more tolerable on the return, daytime trip. Since separate tickets in that direction seem really risky to me, I'd consider a multi-city ticket into one of the European gateways with non-stops from Denver and out of Palermo or Catania. I'd then add on a separate ticket from my inbound gateway to Palermo.

A quick search on google flights came up with a number of 1 stop connections that will get you to PMO. I was able to find connections via MUC, FRA, ZRH, LHR and CDG. You just need to check every day of the week to see which airlines have the connections. The connections through ZRH and CDG are on Edelweiss and Air France which do not fly daily from Denver.

Note - my search was for the middle of June. I'm sure there are fewer options in March and some of the other months.

Edit - If you are flying in March, there seems to be 1 good connection via MUC but only on Saturday departures. Good luck.