Step 1

Combine sugar, cream of tartar, and 1/4 cup water in a medium saucepan over medium heat, stirring until sugar dissolves. Cook, without stirring and occasionally swirling pan for even cooking, until mixture turns a dark amber color. Remove from heat; slowly add apple cider, cream, vinegar, vanilla, and salt (mixture will bubble vigorously), whisking to combine. Return pan to medium heat and bring to a vigorous boil. Reduce heat to a simmer and cook, stirring to dissolve any caramel bits, for 1 minute. Remove from heat and let cool. Set aside. DO AHEAD: Caramel sauce can be made 1 week ahead. Cover and chill. Rewarm slightly before using.

Step 2

Arrange a rack in middle of oven and preheat to 350°F. Generously coat Bundt pan with nonstick spray. Place butter in a medium saucepan over medium heat. Scrape in seeds from vanilla bean; add bean. Cook, stirring often, until butter foams, then browns (do not burn), about 5 minutes. Carefully remove vanilla bean; pour brown butter into a medium bowl and chill until slightly firm, about 45 minutes.

Step 3

Pulse flour and almonds in a food processor until almonds are finely ground. Transfer to a large bowl. Whisk in baking powder and next 7 ingredients; set aside.

Step 4

Using an electric mixer on medium speed, beat brown butter, brown sugar, sugar, and 3/4 cup powdered sugar in a large bowl, frequently scraping down sides and bottom of bowl and beaters, until light and fluffy, about 5 minutes. Mix in lemon zest. Add eggs and yolks one at a time, beating to blend between additions; beat mixture until fluffy, about 3 minutes.

Step 5

Reduce speed to low. Add half of dry ingredients; mix until almost blended. Add milk and mix until almost blended. Add remaining dry ingredients; mix until batter is blended and smooth. Scrape into prepared pan. Tap pan gently on counter to even out.

Step 6

Bake until a tester inserted near the center of cake comes out clean and cake is dark golden brown and has begun pulling from sides of pan, 65-70 minutes.

Step 7

Transfer pan to a wire rack. Let cake cool in pan for 25 minutes. Invert cake onto rack and let cool completely, about 1 hour. DO AHEAD: Cake can be made 2 days ahead. Store airtight at room temperature.

Step 8

Dust cake lightly with powdered sugar. Drizzle caramel sauce over cake, allowing it to drip down sides and into grooves and crevices. Serve remaining sauce alongside.