For the Cupcakes:

1. Preheat the oven to 350°F. Line one 12-cup muffin pan with paper liners. Set aside.
2. Whisk together the flour, baking powder, and salt in a medium bowl. Set aside.
3. In a stand mixer fitted with the paddle attachment, beat the butter on medium-high speed until very light in color, about 3 minutes. Add the granulated sugar and continue beating until the mixture is airy, about 3 more minutes. Scrape down the paddle and the sides of the bowl. 
4. Reduce to medium speed and add the eggs one at a time, beating well after each addition.
5. With the mixer still running on medium speed, add the milk and vanilla to the butter and egg mixture and beat until completely combined. Scrape down the sides of the bowl.
6. Add the flour mixture, and mix at low speed until just combined, about 15 seconds.
7. Divide the batter between the prepared muffin cups. Bake for 15 to 18 minutes, or until a toothpick inserted in the center comes out clean.
8. Set the pans on a cooling rack and let cool for 5 minutes. Remove the cupcakes from the pans and let cool completely on the rack before frosting.

For the Frosting:

1. Briefly stir together the granulated sugar and water in a small saucepan. Bring to a boil over medium-high heat.
2. Reduce the heat to low and continue cooking, without stirring, until the mixture turns dark amber in color, about 8 to 10 minutes.
3. While the caramel is cooking, bring the heavy cream to a simmer in a separate small saucepan, about 2 minutes. Remove from the heat and add the vanilla.
4. Once the caramel has attained a dark amber color, immediately remove from the heat and slowly add the warm cream and vanilla mixture, stirring with a wooden spoon or metal whisk until completely smooth. Set aside until cool to the touch, about 25 minutes, stirring occasionally.
5. The caramel will yield about 2/3 cup. Set aside about 1/3 cup for the frosting, reserving the remaining caramel for drizzling on top of the cupcakes.
6. In the bowl of a stand mixer fitted with the paddle attachment, beat the butter and salt together on medium-high speed until fluffy and light in color, about 3 minutes.
7. Reduce the speed to low, and mix in the powdered sugar until completely incorporated. Scrape down the sides of the bowl.
8. Add the cooled 1/3 cup caramel to the butter mixture. Beat on medium-high speed until airy and thoroughly mixed, about 2 minutes.
9. Decorate the cooled cupcakes, drizzle with the remaining caramel, sprinkle with sea salt, and serve immediately.