Start by adding the unsalted butter to a medium saucepan over medium heat. Allow the butter to melt and cook, stirring occasionally. Watch closely as it begins to bubble and turn a light golden brown, which typically takes 5-10 minutes. The key here is to let the butter get slightly caramelized, adding depth of flavor to your muffins. Be sure to use a light-colored saucepan so you can easily monitor the color of the butter. Once the butter reaches a light brown color, remove it from the heat and set it aside to cool slightly.
In a large bowl, whisk together the flour, cornmeal, baking powder, baking soda, and salt.
In a separate bowl, whisk together honey, buttermilk, and eggs. Add the cooled browned butter to the mixture and stir until combined.
Add the wet ingredients to the dry ingredients and stir gently until just combined. Be careful not to overmix.
Grease a muffin tin with non-stick spray or butter. Divide the batter evenly among the muffin cups, filling each about 2/3 full.
Preheat the oven to 400°F (200°C). Bake the muffins for 12-15 minutes, or until golden brown and set. Test with a toothpick—if it comes out clean, the muffins are done.
Let the muffins cool in the tin for 5-10 minutes before transferring them to a cooling rack.