Step 1

Place a rack in upper third of oven; preheat to 350°. Generously coat pan with nonstick spray or butter, then dust with flour, tapping out excess. Whisk salt, baking soda, and 3 cups flour in a medium bowl to combine.

Step 2

Beat butter in a stand mixer fitted with the whisk attachment on high speed until light and smooth, about 1 minute. With motor running, gradually stream in granulated sugar and beat until mixture is very pale yellow and fluffy, about 5 minutes.

Step 3

Add eggs one at a time, beating well after each addition and scraping down sides of bowl with a rubber spatula as needed. Reduce mixer speed to low and, with motor running, gradually add flour mixture in 2 additions, beating just until incorporated after each addition. Beat in sour cream, oil, and vanilla, scraping down sides of bowl as needed, until combined (be careful not to overmix the batter).

Step 4

Whisk brown sugar, butter, flour, cinnamon, and vanilla in a small bowl to combine. (This is your cinnamon mixture for swirling.)

Step 5

Scrape one third of batter into prepared pan. Drizzle half of cinnamon mixture over. Using a butter knife or skewer, swirl cinnamon mixture through batter. Top with half of remaining batter; drizzle remaining cinnamon mixture over and swirl together again. Scrape remaining cake batter into pan and smooth surface.

Step 6

Bake cake until a tester inserted into the center comes out with only a few moist crumbs attached, 80–90 minutes. Transfer pan to a wire rack and let cake cool in pan 10 minutes. Turn out cake onto rack (use a small offset spatula to loosen if needed); let cool completely.

Step 7

Beat cream cheese and butter in the clean bowl of stand mixer fitted with clean whisk attachment on medium-high speed until combined, about 2 minutes.

Step 8

Reduce speed to medium and gradually add powdered sugar in 2 batches, scraping down sides of bowl as needed. Continue to beat until powdered sugar is incorporated, then increase speed to medium-high and stream in milk and vanilla. Beat until icing is smooth and pourable, about 2 minutes.

Step 9

Set rack with cooled cake inside a rimmed baking sheet and drizzle icing over cake, letting it drip down the sides. Transfer to the refrigerator and chill until icing is set, about 15 minutes.

Step 10

Transfer cake to a large plate to serve.Do Ahead: Cake can be made 3 days ahead. Transfer to an airtight container; cover and chill. Bring to room temperature before serving.