Keep the eggs and buttermilk near 70°F before mixing. Grate 1 large Granny Smith apple and 2 medium carrots on the fine holes. Lightly squeeze the apple for about 20 seconds so it's damp but not dry. Then whisk the eggs, light brown sugar, granulated sugar, refined avocado oil, buttermilk and 1 teaspoon vanilla for 60 seconds. Fold in the grated apple and carrots with broad strokes. The mixture should look glossy and thick with fine orange and pale-green flecks suspended throughout. You shouldn't see pools of oil around the edges. Stop mixing after about 90 seconds total. Keep going too long and the vegetables release more moisture while the eggs get overly aerated. Gentle mixing keeps the crumb moist without making it heavy. Preheat the oven to 350°F now so it's fully hot by the time the pan goes in.

Whisk the all-purpose flour, baking powder, baking soda, 1 1/2 teaspoons ground cinnamon, ground ginger, ground nutmeg and 1/2 teaspoon kosher salt together first. Then add the dry mixture to the wet mixture in two additions. Fold gently for 45 to 60 seconds with wide strokes. Scrape from the bottom after every three folds. Fall Desserts for People Who Hate Pumpkin stay tender when you stop as soon as the flour disappears. The batter should become thick and scoopable with tiny carrot and apple flecks still visible. A few small flour streaks can remain after the first addition but none should remain at the end. Don't mix beyond about 60 seconds after the last flour addition. Overmixing develops gluten and produces a springy cake instead of a soft crumb. Short folding limits gluten formation.

Heat the oven to 350°F and coat a 9-inch square cake pan with 1 teaspoon refined avocado oil. Spread the batter into the pan within 5 minutes of finishing the fold. Push batter gently toward the corners and leave the top slightly rippled instead of perfectly flat. Tap the filled pan twice to release oversized air pockets. Don't let the mixed batter stand longer than 10 minutes before baking. The batter should fill the pan in one thick layer that's about 1 inch deep. Fine carrot and apple flecks should remain visible across the top. Large bubbles should disappear after the two taps but tiny bubbles are fine. Waiting too long lets the baking soda react before oven heat sets the crumb. Fast transfer preserves the batter's lift.

Bake at 350°F for 28 to 33 minutes on the center rack. Rotate the pan 180 degrees at minute 20 if your oven browns unevenly. Check the center at minute 28 with a thin skewer. It should show a few moist crumbs but no wet batter. Fall Desserts for People Who Hate Pumpkin need gentle cake heat so the apple moisture evaporates without drying the edges. Cool for 45 minutes before frosting. The surface should deepen from beige-gold to warm amber and pull slightly from the pan edges. You'll smell toasted cinnamon and apple near minute 25. Press the center lightly and it should spring back without leaving a dent. (I tried cutting it warm once, the crumb compressed under the frosting. Forty-five minutes fixes it.) Go past 33 minutes and the corners can turn dry. Moderate heat sets the starch before the fruit moisture escapes.

Cook the unsalted butter over medium-low heat around 4 on a 10-dial for 4 to 6 minutes. Swirl every 30 seconds until amber milk solids appear and the butter smells nutty. Cool it for 15 minutes to about 80°F. Then beat it with softened cream cheese for 90 seconds. Add powdered sugar, 1 teaspoon vanilla and 1/4 teaspoon kosher salt. Beat another 2 minutes until thick and smooth. Spread the frosting over the completely cooled cake in uneven broad swirls and dust with the remaining 1/2 teaspoon cinnamon. Cut into 9 generous squares after 10 minutes. The frosting should hold soft ridges without sliding down the sides. Fall Desserts for People Who Hate Pumpkin get their richest finish from those toasted butter solids. Frost too early and the cream cheese melts into the crumb. Cooling preserves the frosting's structure and tang.