Is It Possible To Fix Gum Recession Between my Two Front Teeth Which Got Worse During Braces?

Without question your case will be challenging. A tissue graft in this area, for even a very skilled surgeon, will be difficult. Porcelain veneers alone would look too wide. I would find the best periodontist in your area and see his thoughts on tissue grafting this area. If he can improve the area by even fifty percent, then porcelain veneers can be placed for a nice result. Though there is no guarantee this would turn out perfect, I feel it is your best chance. With the right surgeon and cosmetic dentist you should end up with a nice result.

There are several methods to correct loss of gum tissue including graftes.  A bone graft may be necessary prior to soft tissue grafting.  An easier solution that might work well is to narrow the front incisors in the midline and then pull them closer together which might "grow" the tissue back.

It is also possible to grow bone by supererupting teeth orthodontically and then reshaping the teeth.

Another solution is veneers or crowns with pink ceramic to match the gums.

There are other methods that can be used as well but those would solve most problems. 

There are papilla type gum drafts that can be done BUT they only work in specific situations and may not work for you. The problem is that for some reason, you lost bone between the teeth. The only way to have bone grow down is to orthodontically pull the teeth down which will pull the bone down with them and then place new crowns over the teeth to close the gap. It's an aggressive type of treatment but sadly nothing else will work for you. 

This is indeed a challenging situation.  But there is hope.  As the previous doctor recommended, there are various types of grafts.  Bringing the teeth a little closer together by reshaping the teeth between the two centrals.  Also bonding composite may help reshape the teeth or veneers.  Probably the best approach is a combination of these solutions.  If your periodontist is uncomfortable with the grafting you may want to check with a couple others for 2nd opinions.  Dental materials are changing so fast and new techniques give much promise.  Don't give up on this.