For the cake:
Preheat the oven to 350 degrees. Spray 3 9inch round cake pans with nonstick baking spray with flour (like Baker's Joy) or spray with regular nonstick cooking spray and dust lightly with flour. Line bottoms of pans with parchment paper cut to fit; spray with nonstick baking spray with flour (or alternate method above).
In a large bowl, beat the butter & sugar until fluffy, stopping occasionally to scrape sides of bowl. Add eggs & egg yolks one at a time, beating well after the addition of each one. In a separate medium bowl, mix together the milk, sour cream, & vanilla. With mixer on low speed add flour & milk/sour cream mixture a little at a time, alternating between each one. I like to do this in fourths, such as 1 cup of the flour + ¼ of the milk mixture, repeating until both substances are gone. Mix until just blended.
Divide the cake batter evenly between the 3 prepared cake pans; 3 level cups worth per pan. Smooth the tops with a spatula. Bake at 350 degrees for 30 minutes, or until a toothpick inserted into the center comes out clean. Once cakes are done, cool in pan for 10 minutes. Then remove cakes from pans, and cool completely on a wire rack.
For the butter pecans:
Preheat oven to 350 degrees. On a rimmed baking sheet, place pecan halves & diced butter. Sprinkle with salt. Bake for approximately 12 minutes, stirring occasionally. Remove from oven; cool completely. Chop 1 cup's worth of the pecans into small pieces, and set aside for the frosting.
For the frosting:
In a large heavy saucepan, bring dark brown sugar, butter, heavy cream, syrup, & salt to a boil over medium heat, stirring constantly. Reduce heat to medium-low; cook 5 minutes – stirring occasionally. Remove from heat, and transfer mixture to a large heatproof bowl. Gradually add confectioners sugar, apple cider vinegar, & vanilla – beating at a medium speed with an electric mixer until smooth. Let stand for 10-15 minutes, stirring occasionally. Stir in the chopped 1 cup of pecans along with ¼ cup of milk.
Frost & assemble the cake:
On a cake stand, place on layer; top with ¾ to 1 cup's worth of frosting – spreading with a spatula over the cake layer, leaving a small unfrosted area near the very edge. Repeat with the remaining 2 layers of cake. Spread a thin layer of frosting carefully around the sides of the cake, working in small sections. Let the cake sit for 10-15 minutes to allow the frosting to 'set' a little, this is called a crumb coating and you should not expect your cake to look very pretty as this point. After you've allowed the cake to set for 10-15 minutes, use the rest of the remaining frosting to frost the rest of the cake, working in sections. If you find your frosting has gotten a little too thick to frost, add a little splash of milk and stir well to mix thoroughly. Garnish the cake with the remaining unchopped pecan halves.