Orthodontics can be the best choice to close gaps in front teeth in healthy mouth.
You have beautiful teeth. Orthodontics (Braces or possibly Invisalign) would be the best, long term and most conservative option for you. The gap on either side is too wide to fill with either composite or porcelain veneers, which if done, would make your tooth appear very wide.
Another short term (or a quick fix) option, which may also be cost effective, would be having a very good cosmetic dentist add a bonding/composite veneer just to cover part of the gap that is on the front of the...