Combine the saffron and 3 tablespoons wine in a small bowl. Set aside while you prepare the chicken.
Rinse, dry, and season the chicken parts on both sides with salt and pepper. If your slow cooker insert is approved for stove use, heat the oil over medium-high heat until shimmering.* Sear the chicken in two batches until golden brown, about 4 minutes per side. Remove the chicken from the insert/pan and set aside.
Drain some of the residual fat from the insert/pan, leaving about a tablespoon to cook the aromatics. Add the onions and bay leaves, and sauté until the onions are lightly browned, 5-8 minutes. Add the garlic, oregano, 1 teaspoon kosher salt, and a few grinds of black pepper. Cook, stirring, for about one minute, until fragrant.
Deglaze the insert/pan with the remaining 1/2 cup wine and the reserved saffron-infused wine. Stir for about one minute, and remove from heat.
If using a sauté pan, transfer the mixture to your slow cooker insert. Add the chicken stock and rice, stirring to combine. Return the chicken to the insert, along with any accumulated juices. Cover and cook on the slow cooker's LOW setting for 2-3 hours.
Remove lid and discard the bay leaves. Sprinkle the chicken with the chopped pimentos. If the rice looks very dry, add 1-2 tablespoons of water. Replace lid, and continue cooking on LOW for 10 - 15 minutes.
Make the Warm Asparagus Salad and Serve
In a small bowl, whisk together the mustard, vinegar, lemon juice and zest, shallot, salt, and a few grinds of black pepper. While whisking, drizzle in the extra virgin olive oil until the mixture is emulsified. Taste for seasoning and add additional salt and pepper as needed.
Trim and discard any stalky ends of the asparagus and cut the spears into 1-1/2 inch pieces. Fill a large saucepan about two-thirds full of water and bring to a boil. Add a generous pinch of salt and the lower asparagus spear pieces, reserving the tips. (Adding the tips too soon will cause them to fall apart.) Cook for 4-5 minutes and add the tips, continuing to cook until crisp-tender, about 2-3 minutes. Drain.
Toss the warm asparagus with a few tablespoons of the vinaigrette, the capers, and the chopped parsley. Taste for seasoning, adding additional salt and pepper as needed.
Transfer the chicken and rice to a serving platter and top with warm asparagus salad. Drizzle an additional 1/4 cup of the Lemon-Shallot Vinaigrette over the dish and serve immediately.
*If your slow cooker insert cannot be used on the stove, follow your manufacturer's instructions for in-unit browning/sautéing, or use a large sauté pan and transfer the food to the insert in step 5, as indicated.