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Preheat oven to 350°F (177°C). Spray 9 or 10inch springform pan with nonstick cooking spray. I do not recommend a regular circle or square pan, as the cake may rise above. Set aside.
Set out the cream cheese for the frosting so it may soften as you make the cake batter.
In a large bowl with a handheld or stand mixer fitted with a paddle attachment on medium speed, combine the brown sugar and oil. Beat in the yogurt until fully combined – about 60 seconds. Mixture will be gritty and thick. Add the eggs, one at a time, beating well after each addition. Mix in the vanilla. Set aside.
In a separate bowl, combine the flour, baking soda, cinnamon, nutmeg, and salt. With a spatula, manually stir the dry ingredients into the wet ingredients until just combined and all flour pockets are gone – do not overmix. Fold in the finely shredded carrots and pecan pieces. Pour or spoon batter into prepared springform pan.
Bake cake for 32-38 minutes or until toothpick inserted in the center comes out clean. Do not overbake, which will dry out cake. Check the cake at 30 minutes, then again at 32. My cake took exactly 34 minutes. Allow cake to cool completely before frosting.
beat the softened cream cheese and butter together on medium speed for 2-3 minutes until soft, creamy, and combined thoroughly. Add 2 cups of powdered sugar and beat until thick and combined. Add 2 Tablespoons heavy cream and 2 teaspoons vanilla extract. Beat on medium speed for 2 more minutes. Add more powdered sugar until desired thickness is reached. Add salt to taste. I used 1/8 teaspoon to cut the sweetness.
Store cake in the refrigerator, covered, for up to 5 days.