Preheat oven to 350 °F (175 °C). Grease and flour an 8×8‑inch baking pan; line with parchment for easy removal.

In a bowl, whisk together flour, baking powder, and salt.

In a larger bowl, beat eggs with granulated sugar until pale and slightly thickened, about 2–3 minutes.

Slowly pour in melted butter, mixing until combined.

Stir in milk, brewed coffee, and vanilla extract until smooth.

Gently fold in the dry ingredients until just combined—do not overmix.

Pour batter into the prepared pan and smooth the top.

Bake for 25–30 minutes, or until a toothpick inserted in center comes out with a few moist crumbs.

While the cake bakes:
9. In a small saucepan over medium heat, melt butter and brown sugar, stirring until sugar dissolves.
10. Bring to a gentle boil for 1 minute, then remove from heat and whisk in heavy cream and brewed coffee until glossy and smooth.
11. Let cool slightly until thickened but still pourable.

Once the cake is out of the oven, allow it to rest for 5 minutes.

Poke small holes across the cake surface with a skewer or fork.

Pour warm caramel glaze evenly over the cake, allowing it to soak into the holes.

Let cake cool completely in the pan; for extra gooey texture, serve at room temperature.