1 (¼-ounce) package active dry yeast
¼ c warm water (105–110°F)
2 c plus 2 T all-purpose flour, divided
1 c sugar, divided
½ t salt
½ c plain whole-milk yogurt (preferably Greek-style) at room temperature
1 large egg, warmed in shell in warm water 5 minutes
1½ t grated lemon zest
1 t pure vanilla extract
1¼ sticks unsalted butter, cut into tablespoons and softened, divided
¾ lb firm-ripe peaches, halved and pitted
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, ⅔ 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 tablespoon 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 Tbsp flour. Cover bowl with a towel and let dough rise in a draft-free place at warm room temperature until doubled, 1½ to 2 hours.
Spread remaining 2 tablespoons butter in bottom of a 9-inch square baking pan and sprinkle with remaining ⅓ cup sugar. Cut each peach half into 5 or 6 slices and arrange in a single layer in pan.
Stir dough until flour is incorporated, then spread evenly over peaches. Loosely cover with buttered plastic wrap, then kitchen towel. Let rise in a draft-free place at warm room temperature until almost doubled, about 1½ 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.
Note: Kuchen is best the day it is baked but can be kept, loosely covered, at room temperature 1 day.