Preheat oven to 350°; position rack in center of oven. Butter and flour a 9-in.-wide cake pan (at least 2 in. deep) with removable rim; set aside.
Make cake: select a large stainless steel bowl (at least 10-cup capacity) that can nest comfortably in a large pot. Fill pot halfway with water and bring to a boil over high heat, then reduce heat to a gentle simmer. In bowl, combine eggs and sugar. Set bowl over water; with a handheld mixer, beat eggs and sugar at high speed until pale and thick enough to fall from a spoon in a wide ribbon, about 10 minutes.
Remove bowl from heat. Shake flour through a sieve over egg mixture and fold in gently. Add melted butter and fold in gently until no streaks remain. Scrape batter into prepared pan. Bake on center rack until cake is evenly browned, just begins to pull from pan sides, and springs back when lightly touched in the center, about 40 minutes. Set pan on a cooling rack and let cool at least 10 minutes. Run a thin knife between pan and rim. Remove rim and let cake cool completely.
Make tres leches sauce: In a large pot (at least 6-qt. capacity) over high heat, combine goat milk, sugar, corn syrup, and cinnamon stick. Bring mixture to a boil. Stir in baking soda mixture (sauce will foam up) and reduce heat to medium. Simmer, stirring occasionally, until sauce turns a caramel color and reduces to 3/4 cup, 10 to 12 minutes.
Remove sauce from heat; discard cinnamon stick, and stir in condensed milk and whipping cream. Use warm (see Notes).
With a long, serrated knife, cut cake in half horizontally. Leave bottom half on cake pan bottom. Lift off cake top and set, cut side down, on a flat plate.
Put cake bottom (with pan base) on a wire rack set over a rimmed baking sheet. Poke cake bottom all over with a toothpick, being careful not to poke all the way through. Slowly spoon enough warm tres leches sauce (about 1 cup) over cake bottom to saturate well but not cause it to ooze. Let stand until cool, about 10 minutes.
Make filling: Reserve several raspberries to go on top of the cake, then put remaining fruit in a bowl and mix gently with granulated sugar. Set aside. In a chilled bowl, use a mixer to whip cream until it holds soft peaks and is thick enough to spread. Add vanilla and powdered sugar; mix well.
Scoop about 1 1/3 cups whipped cream onto cake bottom and spread level to edge. Dot with sugared raspberries, pushing them down into cream. Carefully set cake top, cut side down, onto cake bottom and neatly align. Poke top all over with a toothpick as before, then slowly spoon about 1 cup tres leches sauce evenly over cake top to saturate well. Smoothly frost top and sides of cake with remaining whipped cream; transfer to a clean serving plate. Cover cake without touching (invert a large bowl over it) and chill at least 2 hours. Cover and chill raspberries if held longer than 2 hours. Cover and chill remaining tres leches sauce.
Uncover cake and decorate with reserved raspberries. Serve with remaining tres leches sauce.
Note: Nutritional analysis is per serving.