Heat milk, vegetable oil, and sugar in a medium saucepan over medium heat, stirring occasionally, until it's hot, but not at the point of boiling. It should be about 170-180 degrees.
Pour milk mixture into the bowl of a stand mixer with paddle attachment (if you don't have a stand mixer, you can do it by hand- see notes below) and let rest for about 10 minutes to cool a bit (you want it at about 110 degrees). Sprinkle yeast over the top of milk mixture. Let sit for 5 minutes.
Add flour, 1 teaspoon cinnamon, and salt to milk mixture and mix on low for 1 minute, until flour is incorporated. Add vanilla extract and ½ cup grated carrots and use your mixer's dough hook attachment to knead on medium speed until dough is comes together and is smooth, 6-8 minutes.
Transfer dough to a lightly greased mixing bowl, cover with a towel, and place in a warm location. Let dough rise for about an hour, until it doubles in size.
Once dough has risen, punch it down and place on a lightly floured surface. Roll dough out into a rectangle, about 12×16 inches.
Using a knife spread butter over the entire dough rectangle, all the way to the edges. Sprinkle brown sugar, remaining 3 teaspoon (1 Tbsp) cinnamon, nutmeg, ginger, cloves, remaining ½ cup grated carrots, and walnuts over the top.
Roll the dough lengthwise into a tight log, and use a sharp knife or unflavored dental floss to cut the log into 12 equal slices (about 1 ¼" thick). It's OK if you end up with a couple extra! Arrange the rolls in 2 9-inch pie plates that are lightly sprayed with cooking oil or brushed with softened butter.
Cover cinnamon rolls with a towel and place in a warm area to rise for about 45 minutes.
When rolls are almost done rising, preheat oven to 350 degrees.
Bake rolls for about 20-24 minutes, until cooked through and turning golden.
Let cinnamon rolls rest for about 5 minutes before spreading with cream cheese frosting and adding sprinkles, if desired.