In a small bowl, mix the yeast and the water and let sit for about five minutes.

Add the bread flour and stir with a spoon until combined.

Cover the bowl with plastic wrap and let it sit at room temperature for 12 to 24 hours. The sponge should be very bubbly.

Begin about 3 to 3 ½ hours prior to baking time.

Add the water, 1 tablespoon of the oil, the sponge, the yeast, rye flour, and the flour in the bowl of a stand mixer. Mix with a spoon or dough whisk, then mix with the dough hook of your mixer on low for about 2 minutes.

Continue to run the mixer, and add the salt slowly.

Increase the mixer speed to medium, and mix the dough until it is smooth and is beginning to pull away from the bowl. This should take about 6 to 8 minutes.

Lightly oil a large bowl or dough rising bucket with olive oil, and add the dough. Cover with plastic wrap and allow to rise at room temperature until doubled, about 60 to 90 minutes.

Lightly flour a work surface and gently turn the dough out of the bowl.

Pulling from underneath the dough, fold the four 'sides' of the dough to the top center.

Turn the dough over so the folded side is down, place it back into the bowl, cover with plastic wrap, and let it sit at room temperature until doubled, about an hour.

Place ¼ cup of the olive oil (infused or not) into each of the two cake pans and tilt the pans to spread the oil evenly.

Lightly flour the work surface and turn the dough out gently. Be careful not to deflate it. Cut it into two equal pieces and place each piece into an oiled pan.

Gently urge the dough into a round shape with your fingers, and cover the pans with plastic wrap. Let the dough rest for 30 minutes.

While the dough is rising, prepare the chiles and peppers.

Preheat the oven to 400 degrees F and pile the cut chiles in the middle of a rimmed baking sheet. Pour 2 tablespoons of olive oil on top, and sprinkle with a pinch of kosher or sea salt. mix and spread with your hands.

Roast for 4 to six minutes and remove from the oven to cool.

Reduce the oven to 350 degrees F and pile the peppers in the middle of a rimmed baking sheet. Add 2 tablespoons of olive oil, ¼ teaspoon of kosher or sea salt, and toss with your hands until coated. Spread the peppers out on the sheet and bake about 30 minutes. Remove from the oven to cool.

Raise the oven temperature to 450 degrees F and place a rack in the middle of the oven.

Once the dough has rested in the pans for 30 minutes, gently press your fingertips into the rounds 5 times to push the edges out.

Press the pieces of cheese into the dough one at a time while pressing toward the rim of the pan. The cheese should be flush with the dough.

Press the tomatoes into the dough using the same method.

Sprinkle the chiles evenly over the top and gently pat them down.

Press the bell pepper pieces into the dough next, but not as far down as the cheese or tomatoes.

Sprinkle with the thyme and sea salt.

Let the dough rise for about 30 minutes, until puffy.

Bake until golden brown, about 30 to 40 minutes.

Move the pan to the floor of the oven to crisp the bottom, about 2 to 5 minutes.

Remove the pan and move the focaccia to a cooling rack by slipping a fork underneath and sliding it carefully out of the pan.

Let cool a few minutes.

Cut into wedges and eat!