Peaches: Bring 6 cups water, sugar, and chamomile blossoms to a boil in a medium pot, stirring until sugar is dissolved. Boil gently until reduced slightly, about 10 minutes. Remove from heat and let cool 1 hour. Strain out solids, then return liquid to pot and return to a simmer. Add peach halves and cover with a parchment round. Let simmer 3 minutes, then remove from heat and let peaches cool completely in liquid. Refrigerate, covered, until chilled, at least 2 hours and up to 2 days.
Remove peaches from liquid, peel, and cut each half into fifths. Pass liquid through a fine sieve and measure out 1 1/2 cups. Return peaches to remaining liquid. Transfer 1 1/2 cups liquid to a pot; bring to a boil and reduce until syrupy (about 2 tablespoons), about 15 minutes. Let cool completely.
Crusts: Meanwhile, combine butter and sugar with a mixer on medium speed until smooth, about 3 minutes. Add egg yolks, then flour, salt, and tea; mix until dough just starts to come together. Dividing evenly, press dough evenly into bottom and up sides of two 4-by-13-inch rectangular tart pans with removable bottoms. Refrigerate until firm, about 1 hour.
Preheat oven to 375 degrees. Bake crusts until golden brown, pressing down centers of tarts if puffing up, about 20 minutes. Let cool completely.
Filling: Stir together sour cream and sugar in a bowl and spread in bottom of crusts, dividing evenly. Remove peaches from liquid and blot dry. Dividing evenly, arrange peaches over filling; drizzle with reduced syrup. Serve immediately, topped with chamomile flowers.