Instructions

for the cupcakes

 Adjust oven rack to middle position and heat oven to 350 degrees. Line 12-cup muffin tin with paper or foil liners. Using stand mixer fitted with paddle, mix flour, sugar, baking powder, and salt on low speed until combined. Add butter, 1 piece at a time, and mix until mixture resembles coarse sand, about 1 minute. Add eggs, one at a time, and mix until combined. Add milk and vanilla, increase speed to medium, and mix until light, fluffy, and no lumps remain, about 3 minutes.
 Divide batter evenly among prepared muffin cups. Bake until toothpick inserted in center comes out clean, 18 to 20 minutes, rotating muffin tin halfway through baking. Let cupcakes cool in muffin tin on wire rack for 10 minutes. Remove cupcakes from muffin tin and let cool completely on rack, about 1 hour. (Cupcakes can be refrigerated for up to 2 days; bring to room temperature before continuing with recipe.)

for the salted caramel sauce

 Combine sugar, corn syrup, and water in small saucepan. Bring to boil over medium-high heat and cook, without stirring, until mixture is light amber colored, 4 to 6 minutes. Reduce heat to low and continue to cook, swirling saucepan occasionally, until mixture is medium amber and registers 355 to 360 degrees, about 1 minute longer.
 Off heat, carefully stir in cream, butter, vanilla, and salt (mixture will bubble and steam). Return saucepan to medium heat and cook, stirring frequently, until smooth and caramel reaches 240 to 245 degrees, 2 to 4 minutes. Remove from heat and allow bubbles to subside. Carefully measure ¼ cup caramel into heatproof liquid measuring cup and set aside for frosting. Transfer remaining caramel to heatproof bowl and let both cool until just warm to touch, 15 to 20 minutes.
 While caramel is cooling, use paring knife to cut out cone-shaped wedge from top of each cupcake, about 1-inch from cupcake edge and 1-inch-deep into center of cupcake. Discard cones. Fill each cupcake with 2 teaspoons warm caramel sauce; set aside remaining caramel for frosting.

for the frosting

 Using stand mixer fitted with paddle, beat butter, cream, vanilla, and salt on medium-high speed until smooth, about 1 minutes. Reduce speed to medium-low, slowly add sugar, and beat until incorporated and smooth, about 4 minutes. Increase speed to medium-high and beat until frosting is light and fluffy, about 5 minutes. Stop mixer and add ¼ cup caramel to bowl. Beat on medium-high speed until fully incorporated, about 2 minutes.
 Spread or pipe frosting evenly on cupcakes, drizzle with remaining caramel sauce (rewarming sauce as needed to keep fluid), and sprinkle with sea salt. Serve. (Cupcakes can be stored at room temperature for up to 4 hours before serving.)