This claim raises so many questions! It is one of funnier claims on the site.
First, does Bulgaria imports chickens? After all, if they grew all their own, this claim wouldn't be true.
Poultry World says yes.
Bulgaria’s poultry imports are dominated by broiler meat, while export revenues come mainly from duck products. [...] Broiler meat imports accounted for 94% of total poultry meat imports and grew over 10% in volume. Frozen broiler meat imports in 2019 increased by 2.2% to 54,000 mt, but its share in total poultry imports was stable at 50%. Frozen chicken leg quarter imports grew 2% over 2018 and remained Bulgaria’s primary imported poultry product.
Where does Bulgaria get its chickens from?
Poultry World says:
Major suppliers were Poland (40%), Romania (18%) and Hungary (12%).
The USDA mentions Greece as another supplier with an 11% share, but also talks about local production.
Between them, they account for almost all of the broiler chickens sold.
Note in particular, Poland, Romania, Hungary, Greece and Bulgaria are all members of the EU.
Is poultry in EU countries injected with "well-known fast-growing hormones"?
No! The European Commission has had a ban for decades:
In 1981, with Directive 81/602/EEC, the EU prohibited the use of substances having a hormonal action for growth promotion in farm animals. Examples for these kind of growth promoters are oestradiol 17ß, testosterone, progesterone, zeranol, trenbolone acetate and melengestrol acetate (MGA).
This prohibition applies to Member States and imports from third countries alike. The legal instrument in force is Directive 96/22/EC as amended by Directive 2003/74/EC.
So, the answer to the question is: NO! Chickens do not get injected with growth hormones.
I took a side trip to find out if chickens had a preferred side (are they "lateralized"?)
Yes! We've only known about this since 1979. It is induced by light during the embryonic stage.
Given all this, are chickens even asymmetrical?
(Oh man, that feeling when you ask the literature "No-one has even spent serious effort into looking at the lopsidedness of chickens, have they?" and the literature says "Oh, of course they have. I've got your back.")
A 2012 paper investigated the idea that overcrowding chickens might lead to leg deformations, which could be measured by skeletal asymmetries. Such asymmetries ("fluctuating asymmetries" or FA) were assumed to be distributed randomly between the left and right, which allowed them to be compensate for the "directional asymmetries" (or DA) - i.e. when one side was systemically different to the other (which wasn't thought to be due to stress). (So, the authors wanted to known about FA and remove the effect of any DA, but we are only interested in whether DA exists.)
They measured the sizes of toes and lower leg bones (and eyes and beaks etc.) and tried to pick characteristics that didn't have DA.
The DA of the full data set was analyzed as described for the character selection. Although none of the selected characters had shown DA during the character selection, several characters showed DA when all 427 individuals were included. This was likely caused by the increased power of the DA test when performed on this larger sample. Unlike FA, DA is generally not considered an indicator of stress (Klingenberg, 2003) and can lead to biased FA estimates (Van Dongen et al., 1999). Therefore, all measures were corrected for DA by using the unsigned random slopes of the individual regression lines as unbiased estimates of individual absolute FA (Van Dongen et al., 1999).
What does that mean?
It means the paper, unfortunately, didn't describe exactly how asymmetric undeformed, healthy birds are, or in which direction, but yes, chickens are, on average, bigger on one side than the other (and the amount seems to be small.)
So, why did the OP find that only one side of the chicken is sold in the shops?
I don't have an answer to this.
I am dying to know. I even briefly considered some original research: how much it would cost to "air-task" three or four people to photograph some chicken pieces in their local supermarket, and see if there is even an effect to explain here.
My pet conjecture, that I offer without any evidence at all, is that the OP is right: that this is a widely-believed urban legend in Bulgaria, which has led to a consumer preference for right-handed chicken legs, which leads to them being preferentially selected from the display case by picky shoppers, leaving only the left-handed legs remaining for the OP to find when shopping later in the cycle. Alas, the literature didn't have my back when checking this idea.