1. Preheat the oven to 350°. Spray a 10-inch round cake pan with nonstick spray, line the bottom with a parchment round and set aside. Place the beets on a baking sheet and roast until fork-tender, about 50 minutes. Set aside to cool slightly, then peel and purée in a food processor. Set aside to cool.

2. In a medium bowl, sift together the cake flour, cocoa powder and baking powder.

3. In a standing mixer using the paddle attachment, combine the melted butter, turbinado sugar, brown sugar, espresso, eggs and sea salt. Add the sifted flour mixture and mix until combined. Add the beet purée and mix until just combined. Scrape into the prepared cake pan, smooth the top and bake until set and a toothpick inserted in the center comes out clean, 40 to 50 minutes. Remove from the oven and cool completely in the pan.

4. Make the ice cream: In a medium saucepan set over medium heat, combine the cream, ½ cup of the milk, ½ cup of the sugar and the tarragon. Bring just to a boil, stirring often, and turn off the heat.

5. In a large mixing bowl, whisk together the yolks with the remaining ½ cup sugar and the salt. Slowly pour the hot-cream mixture into the yolk mixture, whisking constantly, then return the yolk mixture back to the pot set over medium-low heat. Cook, stirring slowly and constantly with a wooden spoon, until the mixture thickens slightly, 7 to 10 minutes. Strain the mixture through a fine-mesh sieve and place over an ice bath to cool completely. Stir in the remaining ½ cup of milk and process in an ice cream machine according to the instructions.

6. Slice the cake and serve at room temperature with the goat's-milk ice cream.