When I go to the Korean supermarkets in LA, I usually see half an aisle just dedicated to 고추가루 in all kinds of forms (mild to spicy, fine to coarse grind) and colors. I don't know that there's any specific pepper than it is all about how sweet and mildly spicy 고추가루 is supposed to be. You could start from there to make your own by sun drying and crushing whatever peppers are available to you.
If you are looking for a substitute that you can buy at a western store, try crushed red pepper flakes (the kind you see in the shakers at pizza places). You're going to be adjusting for taste/spiciness anyway, so not much of a dealbreaker.
What varities of chili peppers are most commonly used in Italy
Peperoncini (=literally little big peppers)
Grown and used in all Italy
The green ones - Peperoncini verdi
Used in north Italy, made under vinagre, typical Milano's recipe.
They make it in airtight glass jars and open them in December for Christmas (or whenever)
The red ones - Peperoncini rossi
Grown and used most af all in shouth of Italy. They are put to dry, then made in powder, like chili. Can be from medium hot to high hot. Used for seasoning many recipes in all Italy. Many types of red small peppers are used to be dried and for seasoning recipes.
Peperoncini verdi fritti
http://www.cookaround.com/yabbse1/showthread.php?t=8358 (see the photo 1 and 2)
Typical of south Italy.
They are fried in oil, then dried on a cooking paper, then seasoned with salt.
"Pepperoncini" is a contaminated word by a wrong spelling pronunciation of southern Italy, and "friggitelli" (like "puparuolilli do’ sciumm", in the text) is a local/dialectal denomination, not even known in north Italy.
Peperoncini rossi corti ripieni (=red short filled pepper)
Typical of South, filled with tuna and other ingredients.
Peperoni = (literally = big peppers)
Used in all Italy (the two varieties = the shorter are sweeter) - but any color changes the flavor, so the result, depending on the blend color, changes.
They are used most commonly used to make "peperonata", made with onions, oil and tomatoes (some like a ratatouille). With celery in north of Italy. They can be from sweet to hot.
Can be made in oven also, usually filled. Or passed on any fire to burn the skin, then peeled from the burned skin, reduced in large stripes, put into a container, seasoned with garlic, oil and salt and covered with oil. They let rest at least 24 hours for the flavors to blend.
Best Answer
So I found this camera in my pocket and it turns out it makes phone calls, too. Technology! I called the local Cheesecake Factory and the guy on the phone said:
He sounded pretty agitated, it's 7PM here so I probably caught him in the middle of dinner service.
You can buy them canned or roast them yourself. Much of the taste may have come from the sauce, or perhaps some special prep when they are roasted.
That said as pointed out in comments I am also now convinced that they look a bit on the small side. Tomorrow, during off hours, I'll call them back for a confirmation with more detail. We may have to try a corporate phone call and a tweet, too.
PS I've actually had that same dish and I also enjoyed it.