Heat 2 tablespoons oil in a large saute pan or skillet. Add seasoned chicken breasts and cook on medium-high heat until they turn golden brown; flip and cook until second side starts browning. Remove from pan; reserve.
Add remaining two tablespoons oil to hot pan. Saute onions and mushrooms until softened; add garlic, thyme, salt, and pepper, then cook on medium heat until incorporated (one to two more minutes). Deglaze with lemon juice.
When lemon juice cooks off, sprinkle pan with flour, stirring to create a mini-roux (be sure to cook for at least two minutes so sauce will not taste of raw flour). Add mustard, whipping cream, and milk to pan, whisking until thickened.
Simmer chicken in this sauce until tender and cooked through. Serve with sweet baby broccoli.