Preheat the oven to 350°F (175°C). Lightly grease a 24-cup mini muffin pan or line it with greased mini muffin liners.
Place the eggs and granulated sugar in a large mixing bowl. Beat with an electric mixer for about 2 minutes, until the mixture becomes lighter in color and fluffy.
Place the chopped semisweet chocolate and butter in a microwave-safe bowl. Heat in short intervals, stirring frequently, just until melted. Alternatively, melt them gently using a double boiler.
Stir the melted butter and chocolate together until smooth. Allow the mixture to cool slightly so it is warm rather than piping hot.
Slowly add the chocolate mixture to the beaten eggs and sugar. Add the vanilla and mix until evenly combined.
Add the cocoa powder, all-purpose flour, and salt. Using a spatula, gently fold the dry ingredients into the chocolate mixture just until no obvious streaks of flour remain. Avoid overmixing.
Divide the batter among the prepared mini muffin cups, adding approximately 1–2 tablespoons per cavity. Fill each cup no more than two-thirds full.
Bake the Fudgy Brownie Bites for 15–17 minutes. The tops should look set, while the centers should remain moist and fudgy.
Remove the pan from the oven and allow the brownie bites to cool completely before carefully taking them out. Cooling is important because warm brownie bites are delicate and can crumble when removed too early.