Preheat oven to 350 degrees. Spray an 8x8 square pan with non-stick cooking spray. Line the pan with parchment paper. Spray the parchment paper with non-stick cooking spray as well.
In a medium mixing bowl, add in the butterscotch chips.
In a small fry pan, add in the cubed butter. Turn the heat to medium and let the butter melt. Occasionally swirl the butter in the pan. Pay attention to the solids that will start to stick at the bottom. It will take about 10 - 12 minutes to get to this stage. Watch it carefully though.
Once the butter starts to brown up slightly, remove from heat and immediately add in with the butterscotch chips. Let it sit for a couple of minutes to help melt the chips.
Whisk the butter and butterscotch chips together. Mine didn't melt completely, so I placed the bowl in the microwave and heated it for 30 seconds. I whisked again and repeated heating it in the microwave for 30 seconds. Whisking afterwards. If there are still a few small bits, I wouldn't worry about it. They will melt once the brown sugar is added.
Add in the brown sugar. Using a hand mixer, beat until incorporated.
Add in the egg and vanilla and beat on medium for about 1 minute. The mixture will lighten up.
Add in the grated carrots, flour, ground cinnamon and nutmeg. Beat on low until the flour is just incorporated. Don't over-beat the batter.
Pour the batter into the baking pan. Spread out evenly.
Place in the oven and bake for 30 minutes or until done. Place a toothpick in the center of the brownies and if moist crumbs come out, the brownies are done. If wet batter comes out, let it bake a bit longer.