Make the filling: Preheat the oven to 375 degrees F and drizzle a baking dish with olive oil. Arrange the onion wedges in the dish and set the turkey breast on top. Add the thyme sprigs, season the turkey generously with salt and pepper and drizzle with olive oil. Roast until a thermometer inserted into the thickest part of the turkey registers 165 degrees F, about 45 minutes. Let cool, then shred, discarding the skin. (You should have about 4 cups meat.)
Melt the butter in a large cast-iron skillet over medium heat. Add the diced onion, chopped carrots, celery and thyme, 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring, until the vegetables are tender, about 7 minutes. Stir in the mustard, then sprinkle the flour over the vegetables. Cook, stirring, until the flour is toasted, about 2 minutes. Whisk in the broth, milk and shredded turkey and bring to a boil. Reduce the heat to medium low and simmer until slightly thickened, about 3 minutes. Remove from the heat and season with salt and pepper.
Make the topping: Whisk the flour, cornmeal, chives, baking powder, sugar, 3/4 teaspoon salt, the baking soda and a few grinds of pepper in a medium bowl. Using a pastry blender or your fingers, cut the butter into the flour mixture until it resembles coarse crumbs. Add 3/4 cup buttermilk and stir until the dough comes together. Stir in up to 1/4 cup more buttermilk, a little at a time, if the dough feels dry.
Stir the peas into the turkey filling. Using a spoon, drop about 14 mounds of batter on top. Transfer the skillet to the oven and bake until the topping is puffed and golden, about 25 minutes. Let stand 5 minutes before serving.