Wrap the beets individually in foil and roast in a 350° oven until tender—approximately 2 hours. Leave the oven on.
Let the beets cool then peel, cut into chunks, place in a food processor or blender and puree until smooth. Reserve ½ cup of the beet puree for the cake and ¼ cup for the sauce, and save the rest for another use.
Microwave the chocolate in 30-second increments—stirring between each one—until melted.
Whisk the flour and baking soda together in a bowl. In a standing mixer, whisk the sugar, egg and oil on medium-high for approximately 2 minutes. While the mixer is running, add the vanilla extract, then the dry ingredients, then the chocolate and finally the ½ cup of beet puree.
Pour into a 9-inch square baking pan greased with butter or oil and bake at 350° until a toothpick inserted into the middle comes out clean—approximately 20 minutes. Let cool and cut into desired shapes.
For the crème fraîche, mix the yogurt and cream together, cover and let sit out overnight in a warm place. Refrigerate to thicken.
Make the sauce. In a blender or food processor, puree the strawberries with the beet puree. Season to taste with lemon juice and sugar. Adjust the consistency with water, if needed.
To serve, spoon the strawberry-beet sauce onto a plate. Place a slice of cake on the sauce and drizzle with the crème fraîche. Garnish with fresh strawberries and mint.