I had a little insight about Bulgarian squats while doing them the other day. This is my thought process as to why they hurt so bad.
Due to the positioning of this exercise, you are never really in a full hip and knee extended position, so you are constantly contracting your working muscles every rep, never allowing them to relax between reps.
Because of that, you aren't allowing your muscle pump venous return mechanism to function as well as it could. Basically the muscle pump mechanism pumps deoxygenated and metabolic waste product filled blood away through continuous sequences relaxation and contraction of skeletal muscle. What is included in the metabolic waste is lactic acid and hydrogen ions from anaerobic respiration (weight lifting). Lactic acid and hyrogen build up create an acidic environment in the muscle and create that burn you get.
However, your working muscles are never relaxed so not only are they producing even more lactic acid and hydrogen ion byproduct because they are working longer than a typical squat, they are also having a hard time pumping the waste away, so you get that really intense burning feeling due to the building acidity.
Not only that, but the continuous contraction also means you are likely to over saturate your muscle fibers with calcium which causes them to "lock" in position. Lactic acid and hydrogen can also impede on basic muscle contraction is levels build too high as well. This is why Bulgarian squats turn you into a hobbling piece of pukey jelly, I think.