Cook beets in a medium pot of boiling unsalted water until tender, 30–40 minutes, depending on size. Drain and rinse under cold water until cool enough to handle. Cut off stem end, then peel and cut beets into large pieces. Transfer to a blender and add 2 Tbsp. water. Blend, adding water 1 Tbsp. at a time as needed, until a smooth purée forms—it should be the consistency of applesauce. Measure out 1 cup purée (reserve remaining purée for another use).
Preheat oven to 350°. Line bottom of an 8"round cake pan with parchment. Grease with oil, then dust with cocoa powder, tapping out excess.
Whisk almond flour, baking soda, cinnamon, and remaining ½ cup cocoa powder in a medium bowl; set aside.
Heat chocolate and remaining 2 Tbsp. oil in a medium heatproof bowl set over a saucepan of barely simmering water, stirring often, until melted. Remove bowl from heat. Stir in vinegar, vanilla, and reserved 1 cup beet purée until smooth.
Beat eggs, brown sugar, and salt in the large bowl of a stand mixer fitted with the whisk attachment on medium-high speed (or use an electric mixer and large bowl) until more than tripled in volume and mixture holds a ribbon for several seconds when beater is lifted above batter,5–7 minutes. Thoroughly beating the eggs is key to creating an aerated, light crumb and is a critical step when using gluten-free ingredients.
Pour chocolate-beet mixture into egg mixture and beat on medium-low speed until combined. Turn mixer off and gently tip in reserved dry ingredients. Beat on lowest speed, scraping down bowl as needed, until combined.
Transfer batter to prepared pan. Bake cake until a tester inserted into the center comes out clean and the top springs back when gently pressed, 45–50 minutes. Let cool 10 minutes. Carefully run a knife around edges of pan, then invert cake onto a wire rack and let cool.
