Make the starter: Place 1½ cups flour, ¼ teaspoon active dry yeast, and 1 cup of water into a medium bowl and stir until combined well. Cover with plastic wrap and set the starter aside at room temperature overnight or for 8 to 10 hours.

Make the dough: Add 5½ cups plus 1 tablespoon of flour and 2 tablespoons of dry active yeast into the bowl of a stand mixer fitted with a dough hook. Mix on low speed for 1 minute until combined. Stop the mixer. Add the dough starter, eggs, sugar, and olive oil and mix on low for 3 minutes. Gradually add 1 cup of water to the mixture and mix for 3 more minutes until the dough comes together. If the dough feels dry, continue adding up to ¼ cup more water. Add the salt and mix the dough on medium speed for approximately 8 more minutes until it is smooth and does not stick to the bowl. Place a towel over the bowl with the dough and let it proof for 30 minutes. Remove the towel and knead the dough for about 3 minutes with the mixer. Grease a large bowl with about 1 to 2 teaspoons of olive oil. Transfer the dough into the greased bowl and roll the dough in the bowl to coat all of its surface with oil. Cover the bowl with a towel and let it proof for another 2 hours until doubled in size.

Make the caramelized quince: Place the quince into a heavy bottomed pot with the sugar, cardamom, and cinnamon over medium heat. Cook the mixture for 10 minutes, stirring occasionally, until all the sugar is dissolved and the quinces start to release liquid. Reduce the heat to low and continue to gently simmer the fruit for 1½ hours, stirring occasionally, until the quince is tender and caramelized, the liquid has reduced to a thick sauce and the mixture is bright red in color. Remove from the heat and set aside to cool.

Shape the challahs: Lightly grease a clean counter surface with a bit of olive oil. Turn out the proofed dough onto the counter and divide it into six equal balls. Roll the balls on the counter. Cover the dough balls with a towel and let them rest for 15 minutes. Roll each ball into a flat oval (about 11x6 inches) and tightly roll each oval lengthwise into a log. Cover the 6 logs of dough with a towel and let them rest for 10 minutes. Make two 3-stranded challah braids from the six logs of dough (about 20 inches long). Tightly tuck the caramelized quince along the seams of the challahs, 6 pieces of quince per challah. Transfer each challah onto a parchment paper-lined baking sheet. Cover the loaves with a towel and let the challahs rest for 30 to 35 minutes, or until doubled in size.

Preheat the oven to 400 degrees.

Beat 1 egg with ½ teaspoon of water to make an egg wash. Brush the proofed challahs with the egg wash and sprinkle sesame seeds on top. Transfer the challahs into the oven and bake for 20 to 30 minutes until the challah is golden brown and light in weight. Cool the challah on a baking rack.

Serve warm or at room temperature.