Plum Kuchen
['1 (1/4-ounces) package active dry yeast', '1/4 cup warm water (105-110°F)', '2 cups plus 2 tablespoons all-purpose flour, divided', '1 cup sugar, divided', '1/2 teaspoon salt', '1/2 cup plain whole-milk yogurt (preferably Greek-style) at room temperature', '1 large egg, warmed in shell in warm water 5 minutes', '1 1/2 teaspoon grated lemon zest', '1 teaspoon pure vanilla extract', '1 1/4 sticks unsalted butter, cut into tablespoons and softened, divided', '3/4 pound firm-ripe plums (about 4 small), halved and pitted', 'Equipment: a stand mixer fitted with paddle attachment', 'Accompaniment: plain whole-milk yogurt (preferably Greek-style)', 'sweetened']
Stir together yeast and warm water in mixer bowl and let stand until foamy, about 5 minutes. (If mixture doesn't foam, start over with new yeast.)
Add 2 cups flour, 2/3 cup sugar, salt, yogurt, egg, zest, and vanilla to yeast mixture and mix at medium-low speed 1 minute. Beat in 1 stick butter, 1 Tbsp at a time, until incorporated. Beat at medium speed until dough is smooth and shiny, about 5 minutes. (Dough will be very sticky.) Scrape down side of bowl and sprinkle dough with remaining 2 tablespoons flour. Cover bowl with a kitchen towel (not terry cloth) and let dough rise in a draft-free place at warm room temperature until doubled, 1 1/2 to 2 hours.
Spread remaining 2 tablespoons butter in bottom of a 9-inch square baking pan and sprinkle with remaining 1/3 cup sugar. Cut each plum half into 5 or 6 slices and arrange in 1 layer pan.
Stir dough until flour is incorporated, then spread evenly over plums. Loosely cover with buttered plastic wrap, then kitchen towel. Let rise in a draft-free place at warm room temperature until almost doubled, about 1 1/2 hours.
Preheat oven to 375°F with rack in middle.
Bake until kuchen is golden-brown and a wooden pick inserted into center comes out clean, 30 to 35 minutes. Cool in pan 5 minutes, then invert and unmold onto a rack to cool completely.
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