In a large bowl or the bowl of a stand mixer, combine 4 1/4 cups of flour, 2 packets (4 1/2 tsp) of instant dry yeast, and 1/2 tsp salt.
In a separate, heat-proof bowl, combine 1 cup whole milk, 1/2 cup softened butter, and 1/2 cup granulated sugar. Heat for 60-90 seconds in the microwave and stir to combine the ingredients. The butter should be melted, and the mixture should be warm but not hot to the touch (you don't want to kill the yeast!).
Slowly pour the warm milk mixture into the flour mixture and mix on medium speed with a dough hook or by hand. Scrape around the edges with a rubber spatula to make sure all the flour gets incorporated.
Wring out 1 1/2 cups of freshly grated carrots with a paper towel or cheesecloth. You should be able to squeeze out a significant amount of carrot juice, and the carrots should look kind of like sweet potatoes once they're wrung out. You should be left with about 3/4 cup of wrung-out carrots that weigh half as much as before (from about 210g to 105g). Mix them into the dough at a medium speed until they're worked into the dough.
Add in 1 large egg. Mix at low speed at first, then increase to medium speed until a sticky dough ball forms.
Continue to mix the dough on a medium-low speed for 2-3 minutes until the dough becomes more elastic.
Keep mixing the dough and add 1 Tbsp of additional flour at a time until the dough is tacky but no longer sticky (it usually takes me 1-2 additional Tbsp of flour). The best way to test the dough is to tap the dough with your finger. It should be tacky but shouldn't stick to your finger or leave any residue on your finger.
Once the dough reaches this stage, cover your bowl with plastic wrap and let the dough rest for 10-20 minutes to allow the gluten to relax. This will make it easier to roll out.