Preheat the oven to 375 degrees F. Line a 12-cup muffin pan with nonstick liners. Whisk the flour, sugar, cocoa powder, baking powder, baking soda and salt in a large bowl; stir in the chopped chocolate. Whisk the milk, vegetable oil and eggs in a small bowl until smooth; stir into the flour mixture until just combined.
Divide the batter among the muffin cups, filling them two-thirds of the way. Bake until the edges are just set but the centers are still loose, 8 to 10 minutes. Working quickly, drop 1 heaping teaspoon dulce de leche into the center of each muffin. Return to the oven and bake until the batter rises over the dulce de leche and a toothpick inserted into a muffin (but not into the dulce de leche) comes out clean, 10 to 15 minutes.
Finely grate chocolate over the tops of the hot muffins and sprinkle with flaky salt. Let the muffins cool 5 minutes in the pan, then remove to a rack to cool completely.
Photograph by Ryan Dausch