Prepare the Dough: Chill a bowl and dough hook attachment of a heavy-duty electric stand mixer before beginning. Cut cold butter into cubes, and set aside at room temperature. Add flour, milk, sugar, yeast, salt, and eggs to chilled bowl, and stir until combined. Beat on low speed with hook 30 minutes. Add butter, a few pieces at a time, and beat until incorporated, about 5 minutes.

Place Dough on a lightly floured surface, and shape into a 10- x 14-inch rectangle. Fold in thirds, top to bottom (as you would a letter), then in thirds, left to right. Press out any trapped air. Grease top of Dough with vegetable oil, cover, and let stand in a warm place (80°F to 85°F) until doubled in size, 45 minutes to 1 hour. Fold Dough in thirds again, top to bottom, and left to right. Chill 1 hour before rolling and shaping.

Prepare the Apple Filling: Melt butter in a saucepan over medium. Add vanilla, and bring to a boil. Add diced apples, and stir to coat. Add sugar and cinnamon, and cook, stirring often, 5 minutes. Increase heat to medium-high. Add cider and vinegar, and cook, stirring occasionally, until liquid is mostly evaporated, 10 to 15 minutes. Spread mixture on a baking sheet to cool.

Line a baking sheet with parchment paper, and grease paper with vegetable oil. Place chilled Dough on a lightly floured surface, and roll out to a 12- x 7-inch rectangle. Spread apples on Dough leaving a 1⁄2-inch border. Fold Dough in thirds, as you would a letter. Repeat folding once. Roll into a 12- x 7-inch rectangle once more. Cut into 16 squares. Gently place squares on baking sheet.

Preheat deep fryer to 350°F (or pour oil to a depth of 3 inches in a Dutch oven over medium). Cook fritters, in batches, in hot oil until deep golden brown, 2 to 3 minutes per side. Place on a plate lined with paper towels.

Prepare the Glaze: Whisk powdered sugar, whole milk, and vanilla extract together in a bowl until smooth. Dip fritters into Glaze. Place on a wire rack to cool and set.

Prepare the Salted Caramel Sauce: Combine cream and salt in a saucepan over medium-high; heat just until bubbles form around edges. (Do not boil.) Remove from heat, and set aside. Combine sugar, corn syrup, and water in a separate saucepan, and cook over high until sugar is dissolved, swirling pan occasionally. Sprinkle cream of tartar around edges of sugar mixture, and remove from heat. Gradually add cream mixture to sugar mixture, whisking constantly until smooth. Add sour cream, whisking constantly until smooth. Serve Apple Fritters warm topped with Salted Caramel Sauce and ice cream.