1. Make the pie dough: in a medium bowl, whisk the flour and salt to combine. Add the butter cubes, tossing them through the flour until each piece is well coated. Press each piece of butter between your fingers to flatten each piece into a shard. The idea is to only do this to each piece of butter once, leaving the butter in large, flat pieces.
2. Make a well in the center of the mixture, and add the water and toss with your hands to incorporate. Continue to add more water 1 tablespoon at a time and continue to mix just until the dough comes together. As it begins to come together, you can knead it a few times to make sure its fully combined. It’s important not to add too much water to the dough, which should never be sticky - it should hold together easily in a ball but still feel almost dry to the touch. Form the dough into an even disk, and wrap tightly in plastic wrap. Chill for at least 30 minutes before proceeding.
3. On a lightly floured surface, roll out the dough into a rectangle about ½ inch thick, then fold into quarters. Roll out the dough again to ½ inch thick, and fold into quarters again. Wrap the dough in plastic wrap and chill for 30 minutes before proceeding.
4. Repeat step 3, rolling out and folding the dough two more times. Wrap the dough in plastic wrap and chill for 30 minutes while you make the filling.
5. In a medium bowl, mix the got cheese, thyme, rosemary, red pepper flakes, and cream to combine. Spiralize the yellow beets, then the red beets, placing each color in a medium bowl. Drizzle 2 tablespoons of olive oil over each color, and season both with with salt and pepper.
6. On a lightly floured surface, roll out the dough into a rectangle 12x16 inches wide. Cut the dough into 3x4 inch rectangles and transfer to two parchment lined baking sheets (they can be quite close together on the sheet tray). Dock each piece all over with a fork. Egg wash the whole piece of dough.
7. Preheat the oven to 425 degrees Fahrenheit. Spoon 1 ½ tablespoons of the goat cheese mixture into the center of each pastry, then divide the two bowls of beets between them. Transfer the pastries to the oven and bake until the crust is deeply golden brown and the beets are tender, 25-30 minutes. Cool at least 15 minutes before serving.