Heat the oven to 350°F (175°C). Line the bottom of a 9-inch round cake pan (2-inch sides) with a parchment circle.
Prepare the blood oranges: Finely zest 1 fruit to yield about 1 tbsp; cut away peel and white pith from all the oranges, then slice into 0.25-inch rounds, removing seeds. From the trimmings and extra pieces, squeeze 0.25 cup juice; set the zest, 10–12 neat rounds, and the juice aside.
Make the caramel base: In a small saucepan over medium heat, melt the unsalted butter (4 tbsp), then stir in the light brown sugar until glossy and bubbling, 1–2 minutes. Immediately pour into the prepared pan and tilt to coat the bottom. Arrange the reserved orange rounds snugly over the caramel.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and fine sea salt.
In a large bowl, beat the unsalted butter (0.5 cup, softened) with the granulated sugar until pale and fluffy, 2–3 minutes. Beat in the eggs one at a time, then mix in the vanilla extract and the reserved zest.
Add half of the dry ingredients, then mix in the Greek yogurt and 0.25 cup reserved blood orange juice, then add the remaining dry ingredients; mix just until smooth.
Spoon the batter over the fruit in the pan and gently smooth the top.
Bake until golden and a tester inserted in the cake (avoiding the fruit) comes out clean or with a few moist crumbs, 35–45 minutes.
Cool on a rack for 10 minutes. Run a thin knife around the edge, invert onto a serving plate, and peel off the parchment. Spoon any pooled syrup back over the cake.
Slice and serve warm or at room temperature. Add whipped cream (for serving) if desired.