It's not a restaurant, but you may have luck with the Society for Creative Anachronism (the SCA), and Renaissance Festivals, which SCA members frequent or work at.
The SCA is a group that keeps the arts and culture (and food) of pre-17th century Europe alive. Everyone will be in period correct garb (or at least an attempt it). There will be people of varying experience and authenticity at every gathering, but you need to make an effort to be period correct. There are no "tourist" like at a Renaissance Festivals. Many SCA events involve a feast, or at larger/multi-day camping events, there can be 4-5 "camps" which do a feast every night. Just like everything else the food is authentic to pre-17th century Europe (except where health codes make it illegal).
They have public events, which are likely your best source for finding authentic medieval food. Even at public SCA events you'll be expected to dress authentically. There is a high degree of overlap between SCA and the local Renaissance Festival. Sometimes the Renaissance Festival will have a feast celebrating holidays like the solstice, and the food will be cooked authentically there too.
Most Renaissance Festivals require their vendors show plausibility their merchandise could exists in the middle ages, and is inspired by the period. I doubt everyone was walking around with turkey legs everyday, but that's an easy place to start.
If you need help with period correct, head to the Renaissance Festival
Chat up vendors and buy some things. They will be able to point you to both local groups and internet resources.
Also - the food will be good, but simple and under-seasoned by modern standards. Remember, in medieval Europe spices were very expensive and thus used sparingly.
My guess as to why there are no "Authentic medieval restaurants" is because every other restaurant would beat them in taste. Even places like Medieval Times will serve more modern food.
Medieval building does not mean medieval food
Several answers have posted menus which contain
Tomatoes, potatoes, corn and other "New World" food that is indigenous to North America.
Carbonated beer - to keep it carbonated you need taps, which didn't exists until 1785.
Water, which was unsafe to drink in Medieval times
You'll have a great experience eating at any of these places. The building is 1000 years old, and plenty of history happened there. The wait staff will be knowledgable and the food will taste great.
None of that makes the food authentic.
Your best bet is still Renaissance Festivals and the SCA. Both a Renaissance Festivals and a Restaurant are experiential, but the type of experience differs significantly.
A Restaurant focuses on provided great food and drink. A Renaissance Festival focus on preserving the arts of the middle ages.
Le cordon bleu opened in Paris in 1895. Restaurants usually try to hype their "classically trained chef". Renaissance Festivals go to trouble to make sure both the ingredients and cooking methods were available 500 years prior to this.
At restaurants the entertainment is the food and drink, and not having to clean up after. Renaissance Festivals have glass blowers, blacksmiths, and (my favorite) jousting.
Renaissance Festival isn't selling food. They're selling a medieval experience, of which simple, under-seasoned food is a part.
One more edit
If you read the link about Cardo Culinaria in the original post you'll see why people might not flock to an authentic experience. All the 20th century people were too fancy to eat with their hands and wanted forks, didn't like Carob juice because lemonade tasted better, and called the food "interesting" (read bad).
Medieval people ate raw foods, simple foods, and under-seasoned foods out of necessity. The current raw foods/simple foods movement has nothing to do with medieval cooking.