Beat milk, eggs, flour, sugar, and vanilla extract together in a large mixing bowl until the batter is thin and without lumps.
Heat a large skillet over medium heat and grease with 1 tablespoon canola oil. Ladle enough batter into the skillet to cover the cooking surface in a thin layer. Cook until bubbles appear on the surface, 2 to 3 minutes. Flip with a large turner and cook until browned on the other side, about 2 minutes more.
Remove cooked palacinky to a plate and roll into a log.