In a medium bowl, whisk together the milk, egg, melted butter, and vanilla. Add the dry ingredients and a large pinch of salt, and whisk to combine. Scrape batter into a shallow pan, like a pie dish.
Heat a large pan to medium-low heat, or electric griddle to 350°F. When griddle is hot, nestle two slices of bread into the batter and let them soak for 5 seconds, then carefully flip and soak the other side for another 5 seconds
Melt 1/2 tablespoon butter on the griddle and add both slices of bread. Cook until the sides are nicely browned and completely dry to the touch, about 4 minutes. Using a wide spatula, carefully flip the bread.
On the cooked side up, add the shaved slices of Gjetost so they melt while the other side is cooking. Once the bottom is nicely browned remove from pan, cover loosely with foil and place in the oven to keep warm.
Repeat and cook remaining bread. Place on a plate; cut in half, if desired; dust with confectioners sugar; optional drizzle with syrup