Here seems to be a lot of consensus that your chick is an EE. Ultimately it can be impossible to know for sure, but here are they facts:
1. It hatched from a blue egg. There are only 8 breeds that can lay blue eggs.
- Cream Legbar. They were created from several breeds. They got the blue egg gene from the Araucana. They have a barred look from the barred rock, as well as their single 5 point comb. They do not tolerate confinement and can become aggressive if unable to forage. You do not have this breed.
- Arkansas Blue. This is a breed currently being developed by the University of Arkansas. It is just a cross of a White Leghorn and Araucana. It is not yet available to the public, but really anyone could cross the 2 breed. There are genetic issues with the Araucana (more below) that I assume they are trying to eliminate before making the breed available. No idea what the breed will look like, but I highly doubt you have this breed.
- Lushi. This is a breed originating in China. They lay both blue and pink eggs. They are small birds and lay medium sized eggs. While they are growing in popularity, they remain uncommon in the US. They are dark and have a small single comb. You do not have this breed.
- Dongxiang. This is a very valuable chicken in China. It has a fibromelanosis phenotype, making its skin (including the single comb) black. You do not have this breed.
- Whiting True Blue. This breed was developed almost 100 years ago by crossing Araucanas and Leghorns (like the Arkansas Blue). They were bread for 3 features: bright blue eggs, high laying frequency and beautiful heckle feathers (for fly fishing). They are not a recognized breed and come in a variety of colours and colour patterns. They have a pea comb. They are non-aggressive, very energetic and love to free-range. You COULD have this breed.
- Araucana. They are they first blue egg layer in North America. At some point in their history they were infected by a retrovirus that rewrote their DNA resulting in blue eggs. They have tufts and are rumples. Unfortunately the gene that creates the tufts is problematic. If both parents carry the gene, the death rate of unhitched chicks as very high. For this reason they are rare in the US. They have a pea comb but do not come in the colour of your chick. You do not have this breed.
- Ameraucana. They were bread in the US from the Araucana to eliminate the lethal eat tuft gene. In Europe the breed is not recognized and is considered a variation of the Araucana. According to the APA, they have muffs, beards, a tail, a pea comb and blue legs. They come in a variety of colours, including mauve (sometimes called lavendar). They prefer to free-range (what chicken wouldn't really?) but tolerate confinement. They are a friendly, docile and intelligent breed. Mine are terrible flyers. You COULD have this breed.
- Easter Egger. They are not considered a breed. They are a cross between a blue layer and another breed. They come in any number of colours. Their eggs can be any colour. They can have any comb type. They may or may not have muffs or a beard. They are the mutt of the chickens yard (don't think I am a hater as I have several EEs and love them all). They are generally friendly, docile and curious chickens. You COULD have this breed.
In short, the hens who layed your blue egg was almost certainly a Whiting True Blue, Ameraucana or Easter Egger.
2. You stated that your mother in law has a rooster. I take this to mean she has just one rooster. If so, he must be the father. If she knows what breed he is, you know half of the parentage. If she does not know what he is, you should post a picture and we could make some suggestions.
As I said in a previous post, I believe you have a male mauve Ameraucana chick, as it looks just like my guy Tortoise did at that age. All of my mauve Ameraucanas have VERY soft feathers. They have squared ends (with the exception of the cockerles heckle and tail feathers) as can be seen in the first pic. The second pic is what Tortoise looks like now at 5 months.
snow I’ve got how has started to grow ear muffs and is rumples today I found 2 little feathers growing from its bun 
ps I live in Italy.
He is definitely NOT an Ameraucana or Whiting True Blue. He COULD be an Araucana if the muffs grow into tufts, but he is most LIKELY an Easter Egger. Since you live in Italy it doesn't really matter as Ameraucanas and EEs are considered to be the same.
