Hi Karolinka, 
My name is Cvetan (Цветан) and its nice to meet you!
I was doing a google search for "the best burger in Sofia" and I saw your site.
I have had a craving for a great burger for over a year now (since the last time I was in the US - I went to college in VA, also my mom lives in IL, so I visit often) and I have been on something like a "quest" to find a really good, traditional hamburger here in Sofia!
I live in Mladost and it's funny, but I didn't know about this bakery place, however your article intrigued me and I will try and visit it asap.
In the meantime, I thought that I would share my burger findings with you and your readers, I actually have only one decent place so far,  called "бар на края на Вселената" (http://www.vselenata.com). They have one location in downtown Sofia and one in the "Student's town" (they also have one in Mladost, but they don't serve burgers there!!! :D ). 
That burger is 100% beef and they serve it with something, that can be described as an omelette patty and really good wedged, crispy potatoes with honey mustard sauce (they are deeeeeeeelicious :) ). Its not a big portion and it costs like 5.50 lv, but it's pretty much the closest thing to a traditional US burger, that I've had here (not counting the junk at McD's and BK)

Another place (unconfirmed yet, by me personally at least) is a bar called Flannigans, located in the Radisson hotel, right across from the Parliament (Народното Събрание).
A couple of US basketball players (playing for one of the teams here in Sofia) highly recommended the burgers there and from what I see on google - other people seem to be confirming this and I am actually about to check them out this week!

I really am on a "hunt" for a really good burger and if you get any more chance encounters like this one with the "Brown Bakery", I would appreciate hearing about it! I will make sure that I follow this article and if you are interested I will share my experiences as well!

Again - nice to meet you (I actually read some of the other articles on your site, your story is cool) and thanks for the burger info!!