Transit visas are no longer available from AZ embassy in Iran, only a 30-day tourist visa is available. No LOI needed, just 1 passport photocopy, 1 passport photo and a straightforward application form (in English/Azeri) obtained from the consulate.
Consulate is open Sundays, Tuesdays and Thursdays, 10am - 1pm, and 5 - 6pm for passport collection. If all required paperwork is handed in by 1pm, you can normally have the visa same day. Here is what you do:
Go to Ghetariyeh metro station on the northern (red) line, when exiting the station, jump in one of the minivan/shuttle taxis for Chizar (5000 rials). After 10-15mins, once most of the passengers have got off, you are in Chizar. Look for Bank Saderat on your right, where the road curves to the left with shops on both sides of the road; get out here. Take the road left side of Bank Saderat, take a sharp right straight away, then next left past a couple of grocer shops, then another right going downhill (with some more small shops on your left side), and the consulate will be on your left after 200 m or so. Otherwise, if you go to the end of the shuttle taxis line, walk to the opposite side of the 'Emmamdazeh' (shrine/mosque) from where you are dropped off, walk straight down the plaza to the end, turn right, left at the photocopy shop then straight down hill. Contrary to previous posts, this photocopy shop knows nothing about the application forms, but its very straightforward from the embassy, and you can leave the 'Azerbaijan contact' details section empty without problem.
There's little point arriving extra early, just be there by 10am sharp. You have to wait to be called in (30 mins), then you go in to the window where 4 Azeri women (who are minus hejabs and speak good English) look at your passport and give you the application form and a slip of paper with the banks address to deposit the above fees. Go back to the front of the Emmadazeh, take the shuttle taxi back to Ghetariyeh, then a savari from the main road there north towards Qods Square, the bank will be on your left after 1km or so, give the driver the paper and ask him to tell you where to get off, as the sign is only in Farsi. Once there, go straight upstairs, give someone the paper, say deposit for Azerbaijan visa and how much you need to pay (in Farsi). You then need to show your passport and have them painstakingly fill out some paperwork, then you wait for your name to be called (30-45 min again). You must pay in the currency the embassy tell you, if you don't have enough, go back south down the main road from the bank about 500m and you'll find a money changer on the right side, with rates posted in English.
Once you paid, you get the important deposit slip, savari/shuttle taxi back to Ghetariyeh/Chizar, and go back to the consulate and queue by the door. A bit of push/shove to get in front is necessary, as by now there's lots of people there (another 45mins wait). Then you hand all paperwork over to the woman (completed/signed application form, 1 x passport copy/photo, deposit slip), and then if all is good they'll tell you to come back at 5pm, where you push/shove/line up again at the door (20-30 min wait) and get your visa. Single entry, 30 days, valid for 2 months from date of issue.