Preheat the oven to 425 F. Line a baking sheet with parchment paper.
In a medium saucepan add the farro, onion, bay leaf, 2 teaspoons of salt and enought water to cover by 2-inches. Bring to a boil over high heat and then reduce heat to low and simmer for 25-30 minutes uncovered. Reserve about 1 cup of the cooking liquid, then drain. Discard the onion and bay leaf.
In a mini food processor, pulse the bread to coarse crumbs. Transfer the bread crumbs to the parchment lined baking sheet. Drizzle with 1 tablespoons extra virgin olive oil and a pinch of kosher salt. Toss to combine and spread into an even layer. Bake for about 8-10 minutes, stirring half way through. Set aside to cool. Keep the oven on.
In a 12-inch skillet over medium-high heat add 2 tablespoons extra virgin olive oil. Add the mushrooms and shallots and sauté until tender and golden brown, about 5 minutes. About one minute before the mushrooms are tender add the garlic and caraway and sauté.
Add the red wine and a pinch of salt and bring the wine to a simmer. Stir in the cooked farro, and 3/4 cup of the cooking liquid.Add the thyme and cook until mixture is thickened and liquid absorbed, about 2 to 3 minutes.
Transfer to a serving baking dish if desired. Make 4 shallow wells in the mushroom farro mixture. Crack an egg into each well. Season the eggs with salt and pepper. Transfer to the oven and make for about 6-10 minutes. Ideally you want the whites cooked and the yolks runny.
To serve; sprinkle with chives and rye bread crumbs.