I glanced at this profile scrolling through and immediately recognized it, even though it’s been over 20 years since I did it. That was an epic day - 129 miles with 16,000 feet elevation gain, most of the peaks at about 8500 feet.
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I have not had the opportunity to climb in the Alps, so apologies if I did not understand the assignment ;-), but if you are primarily looking for extended climbs in a beautiful setting, I’ll mention a few that I didn’t see in the prior responses.
Pikes Peak (CO) – this is an epic climb; very challenging due to elevation
Haleakala (HI) – I haven’t done it yet, but it’s one of the biggest climbs in the world
and in the East, smaller scale but still challenging:
Skyline Drive, Front Royal gate to Hogback summit (VA)
Reddish Knob (WV) – there are road routes and a challenging gravel route, which have been featured in the Alpine Loop Gran Fondo
Spruce Knob (WV – Climbing Spruce Knob , WV by bike - cycling data and info is one route, and there’s a fun gravel race called GRUSK that incorporates this mountain)
I’ll second Mount Mitchell in NC as well. Great climb, and there is a ton of world class cycling around Asheville.
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Whiteface mountain in upstate NY is similar to the alpe d’huez in length and elevation gain. There’s a nice building at the top too. Lake Placid region has a lot of nice riding too.
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You can explore all sorts of climbs with this site.
I never did any climbing in the US, but I am familiar with the alps and pre-alps (as I have lived in Bavaria for many years). I have also lived in Japan near the “Japanese alps”.
That being said, “the alps” have quite varied terrain and it isn’t clear to me what that would mean. I know what I like when it comes to climbing, and that can be found in the alps for sure. But that’s usually a combination of scenery, lack of traffic and roads that are nice to ride.
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As a Brit, specifically a Southern Brit; these elevation figures blow my mind. The highest point we have around here is about 230m above sea-level 
Last summer in interior BC, I did 6 different rides that had single climbs of at least 1100m, all within about a 2.5 hour drive of where I was staying. Most aren’t averaging Alps gradients, but serious gains.
Hedley → Apex (21km, ~1420m) This is gravel, and not smooth packed gravel either!
Penticton → Apex (30km, ~1400m)
Revelstoke → Mt. Revelstoke (26km, ~1380m)
Peachland → Pennask (35km, ~1370m)
Oliver → Mt Baldy (37km, ~1365m) Largely gravel.
Vernon → SilverStar (20km, ~1150m)
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I suggest Colorado’s Rocky Mountains, particularly areas like Aspen, Independence Pass, or Mount Evans, as they offer long climbs with gradients similar to the Alps.
Another thing I recommend is making sure you have internet access and not relying on roaming – I made that mistake, and a few hundred dollars flew out of my pocket.
I recommend an eSIM, I’ve used this USA eSIM for a few trips, and it works very well. The cheapest plan starts at just $5.
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California has so much climbing, and you can enjoy it all year round.
I had about 700,000’ of climbing last year WITHOUT a climbing goal.
Never been myself but I met the then senior MTB champion about 15 years ago and he’s from there. The pics he posts on Strava look amazing 
The Appalachian Gap? The Blue Ridge Parkway? The Colorado (and other states) Rockies? Pikes Peak? Mount Rainier and other mountains around Washington State? Some of the Hawaiian Islands? (There’s a huge climb about 30 minutes from here that makes most riders cry because of the ridiculous effort to get to the top, and yet some are able to ride it over and over again 
. Is it ‘Alps like’? Well, again, to some).
I’ve never been to the US but on Rouvy Mt Lemmon came pretty close, no hairpins (not all Alpe climbs have hairpins either) but still a very long steady and quite steep climb.(a few months ago Rouvy had a US climb challenge, I forgot the other climbs but I still remember Lemmon, it was tough)