1 tablespoon canola oil
4 (4-ounce) bone-in, skin-on chicken thighs
1 ½ teaspoons kosher salt, divided, plus more to taste
1 teaspoon black pepper, divided
½ cup chopped bacon or prosciutto (about 4 bacon slices)
1 medium garlic clove, thinly sliced (about 1 teaspoon)
1 small (5-ounce) yellow onion, thinly sliced (about 1 cup)
2 large celery stalks (about 4 ounces), cut into small cubes (about 1/2 cup)
1 medium (3-ounce) carrot, cut into small cubes (about 1/2 cup)
1 cup thinly sliced fresh shiitake mushroom caps (about 2 ounces)
1 cup dry red wine
Pinch of crushed red pepper
1 ¾ cups canned whole peeled San Marzano tomatoes (from 1 [28-ounce] can), hand crushed
4 (3-inch) thyme sprigs
2 cups chicken broth
Braised greens, rice, polenta, or potatoes, for serving
Heat oil in a large Dutch oven over medium-high until oil shimmers. Meanwhile, arrange chicken thighs in a single layer on a baking sheet or plate; sprinkle tops with 1/2 teaspoon salt and 1/2 teaspoon pepper. Flip thighs; sprinkle bottoms with 1/2 teaspoon salt and remaining 1/2 teaspoon pepper.
Reduce heat under shimmering oil to medium. Using a pair of metal tongs, carefully add chicken, skin side down, to Dutch oven. Sear chicken, resisting the temptation to move or turn the pieces until they have browned on the first side, 5 to 8 minutes. Turn chicken over, and cook until browned on other side, 5 to 8 minutes. Transfer chicken to a plate, and set aside. Do not wipe Dutch oven clean.
Reduce heat under Dutch oven to medium-low, and add bacon and garlic; cook, stirring constantly, until aromatic and bacon is slightly crispy, about 2 minutes. Increase heat to medium-high. Add onion, celery, carrot, mushroom caps, and 1/4 teaspoon salt; cook, stirring occasionally, until onion is translucent, 3 to 5 minutes. Add wine; cook, stirring occasionally, until liquid reduces by half, about 5 minutes. Sprinkle with crushed red pepper and remaining 1/4 teaspoon salt; stir in crushed tomatoes and their juices. Bring to a simmer over medium. Reduce heat to low; simmer, stirring occasionally, until sauce starts to get jammy, 5 to 8 minutes. Stir in thyme sprigs.
Arrange chicken thighs in a single layer in mixture in Dutch oven, and add broth. Bring to a simmer over medium. Cover and simmer over low, undisturbed, until chicken and vegetables are very tender and a thermometer inserted in thickest portion of chicken registers at least 165°F (temperature will likely be around 190°F), 40 to 50 minutes. Using a wooden spoon, break up some of the tomatoes.
Remove Dutch oven from heat. Let chicken mixture rest, covered, about 10 minutes. Remove and discard thyme sprigs. Taste tomato sauce, and add additional salt to taste. Serve with braised greens, rice, polenta, or potatoes, if desired.
