Heat the oven to 350°F. Grease three 8-inch round cake pans at least 2 inches deep. Line the bottoms with parchment, grease the parchment, and lightly flour the pans.

Whisk together the flour, baking powder, and salt. In a separate large bowl, beat the softened butter and sugar for 3–4 minutes until pale and fluffy, scraping the bowl as needed.

Beat in the eggs one at a time, followed by the vanilla. On low speed, add the flour mixture in three additions, alternating with the milk in two additions. Begin and end with flour; mix only until incorporated, then scrape and fold briefly to catch unmixed batter.

Divide the batter evenly among the pans. Bake for about 25–30 minutes, checking from 23 minutes. The tops should spring back lightly and a toothpick should have a few moist crumbs but no wet batter.

Cool in the pans for 10 minutes, then turn onto racks and remove the parchment. Let cool completely, about 60–90 minutes more as needed. Have the cooled layers, cake plate, and offset spatula ready before finishing the icing.

In a saucepan, gently heat 2½ cups of the icing sugar with the evaporated milk, butter, and salt, stirring until dissolved. Keep warm without boiling.

Combine the remaining ½ cup sugar and the water in a heavy, light-colored pot with at least 4-quart capacity. Stir to moisten, then cook over medium heat without stirring until amber. Swirl the pot gently if needed. Remove from the heat before the caramel becomes very dark or smells burnt.

Carefully add the warm milk mixture in several additions, keeping hands and face back from the hot steam and bubbling. Return to medium-low heat and stir until any hardened caramel dissolves.

Cook, stirring steadily across the bottom to prevent scorching, until a candy thermometer reads 237°F near sea level. Keep its tip in the icing without touching the pot. This stage may take about 15–25 minutes; use temperature rather than the clock.

Remove from the heat, stir in vanilla, and let cool undisturbed to about 180°F. Beat with a hand mixer on low, then medium, for roughly 5–12 minutes, until thick enough to spread and a ribbon briefly holds its shape. Avoid splashing the hot icing.

Spread about ½ cup icing between each cake layer. Immediately cover the top and sides with the remaining icing. If it stiffens before you finish, stir in the optional warm milk 1 teaspoon at a time until spreadable. Let the cake set for about 1 hour before slicing into 16 portions.