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For pear compote:.
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Mix sugar, lime juice, and salt in heavy large saucepan.
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Add pears and toss gently to coat.
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Cover and cook over medium-low heat until pears are just tender, stirring occasionally, 10 to 12 minutes.
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Transfer mixture to bowl.
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DO AHEAD: Can be made 1 day ahead.
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Chill until cold, then cover and keep chilled.
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For buttermilk spice cake:.
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Preheat oven to 350F Butter and flour 9-inch-diameter cake pan with 2-inch-high sides; line pan with round of parchment paper.
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Sift first 9 ingredients into medium bowl.
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Using electric mixer, beat butter in large bowl until fluffy.
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Gradually add sugar, beating until smooth.
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Beat in eggs 1 at a time, beating to blend between additions.
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Scrape in seeds from vanilla bean and add lime peel; beat to blend.
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Beat in flour mixture in 4 additions alternately with buttermilk in 3 additions, scraping down bowl occasionally.
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Transfer batter to prepared pan.
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Bake cake until beginning to brown on top and tester inserted into center comes out clean, about 30 minutes.
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Cool cake in pan on cooling rack.
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DO AHEAD: Cake can be made 1 day ahead.
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Cover and let stand at room temperature.
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Cut around pan sides to loosen cake.
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Turn cake out onto rack; peel off parchment and turn right side up onto platter.
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Sift powdered sugar over.
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Cut into wedges.
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Serve with pear compote and dollop of creme fraiche.