I don't advise ever going anywhere with only one Epi Pen. The Epi Pen only buys you time until paramedics can arrive, so if they take longer than 15 min to get to you, you may need another dose of epi. Also, what if one pen malfunctions or something else happens to one of the pens? Better safe than sorry and always have two.
I bring 6 Epi Pens for my DD when we travel. I'm sure it's overkill, but it's what makes me feel comfortable, so that's what I do. Once you decide on how many to bring, be sure to pack all of them (as well as Benadryl and any other meds your DD may need) in your carryon. Never put those meds in your checked luggage.
As others have mentioned, you need to ask for the ingredient book at CS places. Sometimes this will result in you having to wait for a manager to talk to you. Other times, they will just give you the book (this is one area where there seems to be no consistency). Because you may have to wait for a manager, try to get to CS places before your DD is really hungry (we learned this the hard way). We had a really long wait a couple of times and the kids were starving and miserable.
At TS places, the chef will come to your table and go over buffet or menu choices with you. Sometimes the bread on the table is safe, but most often, it's not. We've been to a few places where the chef made DD some allergy safe rolls since she couldn't have the regular dinner rolls. Also, some places have given her some crackers to munch on while the rest of us are having the bread. Baked goods (both breakfast items and desserts) are tough. Most of them will be off limits due to ingredients or cross contamination. However, Tusker House has Krispy Kreme donuts on their buffet and they're safe, so we always make an ADR there. There were a few places where the desserts were made in house and DD could have them, but at most places she ended up with ice cream.
We always bring hand wipes so that DD can clean her hands before she eats. Between railings, ride lap bars, door handles, etc. you have no idea what you are touching and what allergens might be lurking there, so making sure hands are clean before eating can minimize risk.
We also bring safe snacks to the parks, but we seldom need them. We always seem to be able to find something for DD. We've even done the Wishes Dessert Party twice and both times they were able to provide safe desserts for her.
Nut allergies can be really scary at first, but you'll be fine! We've been dealing with DD's for 10 years now and Disney is the easiest vacation spot when dealing with food allergies.
Have fun!