Place chicken pieces in a single layer on a large platter and season with salt and pepper.
Rest at room temperature for 15 minutes to allow skin to dry.
Heat oven to 250°F.
Melt butter in a large skillet over medium heat. Add chicken, skin side down. Cook for 5 minutes, or until browned. Remove chicken from pan.
In same skillet, sauté yellow onion, leeks, pearl onions, and garlic over medium heat until fragrant.
Add flour, reduce heat to low, and stir until mixture is light brown, about 2 minutes.
Add white wine and broth. Stir to ensure all clumps of flour are broken up.
Return chicken to pan and add bay leaf and thyme. Cook for 30 minutes, or until chicken reaches 155°F. (The chicken will continue to cook and reach the desired temperature of 165°F as the sauce cooks.)
Remove chicken from pan, wrap in foil, and place in oven to keep warm.
Bring remaining mixture in skillet to a boil over medium heat. Meanwhile, whisk together mascarpone, mustard, and egg yolks.
Remove skillet from heat and slowly pour egg mixture into skillet, whisking constantly to avoid scrambling eggs.
Remove chicken from oven and arrange on a large platter.
Pour sauce over chicken and serve immediately.