Chop roasted and skinned beets and puree beats in a blender, adding water 1-2 tablespoons at a time until mixture is completely smooth. Measure out 1.5 cups and set aside.
In a medium bowl, combine water, yeast, and honey; stir until dissolved. In a large bowl, whisk together flour and salt. Add the yeast mixture, 4 tablespoons of olive oil and 1.5 cups of beet puree. Stir until flour is just incorporated. Dough will be very wet, and no kneading is necessary. Cover bowl tightly with plastic wrap. Leave out at room temperature to ferment until at least doubled in volume; 12 -14 hours (rising time will vary considerably depending on the season).
When ready to bake, spread 2-3 tablespoons of olive oil onto an 18-by-13-inch rimmed baking sheet. Using a spatula or your hands, gently fold dough onto itself and transfer onto the prepared pan. Pour an additional 2 tablespoons of olive oil over dough and gently stretch it to the edge of the sheet by placing your hands underneath and pulling outward. The dough will keep shrinking, so repeat and stretch once or twice over the course of 30 minutes until it’s fully stretched out and completely covers the pan.
To dimple the dough, use oiled fingertips and pressing the pads of your first three fingers in at an angle.
To make the brine, stir the salt, water, and vinegar together until the salt has dissolved. Pour the brine over the dough to fill dimples and set aside at room temperature for the final proof, until the dough is light and bubbly, about 45 minutes.
30 minutes into this final proof (15 minutes before baking), adjust oven rack to the center position and preheat oven to 450°F. If you have a baking stone, place it on rack. Otherwise, invert another sturdy baking sheet and place on rack. Allow to preheat with the oven until very hot, before proceeding with baking.
When ready to bake, sprinkle focaccia with flaky salt, to taste (remember there’s also salt in the brine).
Bake for 25-30 minutes, directly on top of stone or inverted pan, until the bottom crust looks golden brown and crispy. To finish browning top crust, place focaccia on upper rack and bake for 5 to 7 minutes more.
Remove from oven and brush with 2-3 more tablespoons of olive oil over the entire surface. Let cool for 5 minutes (the oil will absorb into the bread as it sits). Serve warm or at room temperature.