In the bowl of a stand mixer fitted with the paddle attachment, combine 2 cups flour, sugar, yeast, and salt.
In a medium saucepan, heat 1 cup water, butter, and vanilla over medium heat until butter is melted and an instant-read thermometer registers 120° to 130°. Add warm butter mixture to flour mixture, and beat at medium speed until combined and cooled slightly, about 2 minutes. Add egg and egg yolk, and beat at medium speed until combined. With mixer on low speed, gradually add remaining 2½ cups flour, beating just until combined and stopping to scrape sides of bowl.
Switch to the dough hook attachment. Beat at low speed until a soft dough forms, about 8 minutes, stopping to scrape sides of bowl and dough hook. (Dough will be slightly sticky.) Turn out dough onto a lightly floured surface, and shape into a smooth round.
Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°) until doubled in size, about 1 hour.
Recipe by Louisiana Cookin' at https://louisianacookin.com/salted-caramel-chocolate-king-cake/