For the pizza dough: Add 1/2 cup warm water to a small bowl. (If the water is too hot, the yeast will die; if it's too cold, the yeast won't activate). Add the yeast and sugar, stir to combine and let sit for 15 minutes. The water will bubble at the surface and smell very yeasty; this is what you want.
Combine the flour and salt in a large mixing bowl. Make a well in the center and add the olive oil and the yeast/water mixture. Using a fork, gradually stir the flour into the yeast/water. Once most of the flour is combined, turn the dough out onto a smooth, clean, flour-dusted work surface and knead until smooth.
Lightly coat the mixing bowl with oil and return the dough to the bowl. Cover with plastic wrap and set in a warm place until the dough has doubled in size, at least 1 hour.
For the braised chicken thighs: Preheat the oven to 350 degrees F.
Add 2 tablespoons olive oil to an oven-safe, high-sided saucepan over medium heat. When the oil is just starting to shimmer, place the chicken thighs skin-side down in the pan, being careful to not overcrowd. Cook until the skin is golden brown and releases easily from the pan, about 3 minutes. Flip the thighs and cook until golden on the reverse side, another 3 minutes. (The chicken should not be cooked through at this point, just browned.) Remove the chicken from the pan and set aside. If necessary, add another tablespoon of olive oil to the pan. Then add the onions and cook, stirring, until just translucent, about 2 minutes. Stir in the garlic. When the onions and garlic are translucent, add the tomatoes and jalapeno and stir gently to combine. Cook until the tomatoes start to release their juices, 3 to 5 minutes. Add the chicken stock, bring to a boil and then reduce to a simmer.
Return the chicken to the pan, skin-side up, and transfer to the oven. Bake until the chicken comes apart easily, another 20 to 30 minutes. Shred the chicken.
For the pizza: When the dough is ready, portion it into 6 golf ball-size pieces. (If you're not using it right away, wrap each ball individually in plastic and refrigerate until ready to use.) Dust a work surface with flour and roll each piece of dough into a rectangle (or whatever shape you like; I like irregular shapes).
Place a pizza stone in the oven and preheat to 450 degrees F. Dust a pizza peel with cornmeal.
Top each pizza with some braised chicken, about 1 cup of the braising sauce and 1 cup of the Cheddar. Bake until the crust is golden brown and cooked through and the cheese has melted, 8 to 10 minutes. Remove from the oven and garnish with cilantro. Slice and enjoy!