Is this a day trip or over night? As others have said, there is nothing "Euro" within a quick drive from the City (with the caveat that I have never skied in Europe!). However, it certainly is possible to have fun nearby. Nothing in New York strikes me as charming Euro, with the exception of Whiteface/Lake Placid, which is too far for a day trip. In southern Vermont, Stratton or Mount Snow, or in Northern Vermont Stowe. If you have three or more days, you have better options.
1. Good snow - not icy/difficult:
This really varies day to day and hour to hour. Ice is a fact of life in the northeast, though grooming and snowmaking has gotten better. Try not to go after a warm spell. Even if there hasn't been much natural snow, a week or so of sub-freezing temps will have most local places in good shape in terms of coverage. I skied Belleayre over New Years and, while the whole mountain wasn't yet open, coverage was nice on what was open. Parts were icy, though. If you can go on a weekday, you'll have the places to yourselves.
2. large area to have different slopes available
In New York, the principal skiing areas are the Catskills (~2.5 hours from the City) and the Adirondacks (~5 hours). There also are a handful of tiny areas an hour or so away.
Most of the Catskill areas are similar in size. The major areas are Hunter, Windham, and Belleayre. Of those three, I prefer Belleayre, but that's just my opinion. None are huge, but plenty of fun for a day or two. Belleayre and Windham are more family oriented (and have more of an "outdoorsy" vibe). Hunter attracts more of a singles crowd. If you go on a holiday, try Plattekill, though only if there has been some natural snow. Plattekill is smaller, but with similar vertical and nice terrain. It is off the radar, very basic, but a friendly, fun place to go. Rarely any crowds (which usually results in less ice).
The two major areas in the Adirondacks are Whiteface (which has the largest vertical on the east coast) and Gore. If you have 3 days or more, both are great. Whiteface is near Lake Placid, which is a fun town, and probably the most "Euro" on the east coast. Whiteface has the reputation of being "Iceface," though not sure if that reputation is justified. Gore is not quite as large, but has more variety in skiing in my opinion. Gore is a terrific mountain. The Adirondacks are just plain beautiful.
Also in day trip distance to the City are areas in the Bershshires, Catamount, Butternut, and Jiminy Peak.
Southern Vermont areas are also in range. Stratton (has a base village) and Mount Snow have the most amenities, including slopeside lodging. Also Okemo. Bromley is a great mountain too, and more low-key. Killington/Pico have great terrain, though I'm not a fan of the vibe at Killington. I haven't skied Pico, but have heard nice things.
Farther north, northern Vermont and New Hampshire have bigger mountains, with better chance of good snow. Stowe is a charming town with some of the best skiing in the east. For a luxurious weekend, try staying at the Mount Washington Inn and skiing Bretton Woods.
3. Good teaching for her
I can't speak for all of the ski schools, but I know that Don Boyce, who runs the ski school at Belleayre is top notch. Ask for a private lesson with Don or Chris Erickson; you won't be disappointed.
4. Not far from NYC
See above. Closest is Mountain Creek in New Jersey, but overrun by city slickers and snowboarders, especially on weekends.