Preheat and prepare the pan. Preheat your oven to 350°F (175°C) and position the oven rack in the center. Then, grease a Bundt pan with shortening or butter and dust with flour, or use a baking spray that contains flour.
Blend the wet ingredients. In a large bowl, beat the softened butter with an electric mixer until light and fluffy, about 2-3 minutes. Slowly add the sugar and continue beating until creamy. Then, add the eggs one at a time, beating well after each addition. Mix in the sour cream and vanilla extract until fully combined.
Mix dry ingredients. Whisk together the cake flour, baking powder, and salt in a separate bowl.
Combine. Slowly add the dry ingredients to the butter mixture, alternating with the milk. Start and end with the dry ingredients. Mix until just combined, being careful not to overmix.
Bake the cake. Pour the batter into the prepared Bundt pan and smooth the top. Bake for 50-65 minutes until a cake tester or toothpick inserted into the center comes clean.
Cool the cake. Let the cake cool in the pan for 5 minutes, then turn it onto a wire rack to cool completely before icing.
To make the icing:
Caramelize the sugar. In a small, heavy-bottomed saucepan (at least 3 quarts), sprinkle ⅓ cup of granulated sugar evenly across the bottom. Cook over medium heat, stirring constantly, until the sugar melts and turns a light amber color. Be patient, as the sugar may clump before melting smoothly. Keep an eye on it, it will go from golden to burnt quickly.
Prepare the milk mixture. Whisk together the milk, the remaining sugar, and 1 teaspoon of flour in a separate large saucepan. Heat the mixture over medium heat, stirring constantly, until it boils.
Combine and cook. Carefully pour the hot milk mixture into the caramelized sugar. Then, reduce the heat to low and stir until the sugar clumps dissolve completely. Next, raise the heat to medium and cook until the mixture thickens and reaches 238°F on a candy thermometer (soft-ball stage), about 5-10 minutes. Remove the pot from the heat and stir in the butter until melted. Let the mixture sit at room temperature for about 1 hour to cool.
Beat the icing until thickened. Once the caramel mixture has cooled to about 110°F, transfer it to a mixing bowl. Add the vanilla extract and beat with an electric mixer until the icing thickens to your desired consistency. For a thinner glaze, beat for 2-3 minutes. For a thicker consistency suitable for piping, beat for 5-7 minutes. I recommend taking a small amount and putting in on the cake to see how it sets before you pour it all on. I usually end up beating a few minutes more for a thicker texture.
Ice the cake. Once the cake is completely cool, drizzle or pour the caramel icing over the top. If you want, use thicker icing for piping decorative touches.
Notes
If you don’t have cake flour, you can substitute 2 ¾ cups + 1 tablespoon all-purpose flour and 3 tablespoons cornstarch.