In the work bowl of a food processor, combine figs and vanilla extract. With processor running, add ½ cup (120 grams) water in a slow, steady stream until mixture is puréed, stopping to scrape sides of bowl.

Preheat oven to 350°F (180°C).

In the bowl of a stand mixer fitted with the paddle attachment, beat sugars and eggs at medium speed until smooth and slightly thickened, about 2 minutes, stopping to scrape sides of bowl. Add fig purée, oil, and lemon zest; beat until thickened and well combined, 2 to 3 minutes, stopping to scrape sides of bowl.

In a large bowl, whisk together flour, salt, nutmeg, cinnamon, and allspice. In a small bowl, stir together 1 tablespoon (15 grams) warm water and baking soda; add buttermilk and vanilla bean paste, stirring until well combined. With mixer on low speed, gradually add flour mixture to sugar mixture alternately with buttermilk mixture, beginning and ending with flour mixture, beating just until combined after each addition. Fold in chopped pecans.

Spray a 10-inch tube pan with baking spray with flour. Spread batter into prepared pan. Firmly tap pan on a kitchen towel-lined counter several times to settle batter and release any air bubbles.

Bake until a wooden pick inserted near center comes out clean, 55 minutes to 1 hour and 5 minutes, loosely covering with foil during final 5 minutes of baking to prevent excess browning. Let cool in pan on a wire rack for 15 minutes. Invert cake onto a wire rack, and let cool completely.

Pour Caramel Glaze onto cooled cake. Garnish with pecan halves, if desired. Store in an airtight container for up to 3 days.