To make the muffins, preheat the oven to 400°F. Line a 12-cup muffin tin with paper liners or spray with nonstick spray. Cream butter, sugar, and egg in a medium bowl. In a separate bowl, combine flour, baking soda, cinnamon, and salt. Add dry ingredients to the butter mixture, alternating with the pumpkin puree. Mix until just combined. Divide batter evenly among prepared muffin tins. Bake for 18-20 minutes, until a toothpick inserted in the center of a muffin comes out clean. Allow muffins to cool completely before glazing.
To prepare the glaze, combine butter, milk, and brown sugar in a small saucepan over medium heat. Bring to a boil and boil vigorously for 1 minute. Remove from heat and beat in powdered sugar and vanilla. Immediately pour hot glaze over cooled muffins. I used a spoon to pour some glaze on each muffin, but my sister just picks her muffins up and dunks the tops in the glaze. Allow to cool and enjoy!
Recipe by Lulu the Baker at https://luluthebaker.com/pumpkin-muffins-with-caramel-glaze/