In a medium skillet, heat oil to medium high heat.
Season chicken breasts with salt and pepper. Cook on each side until golden brown but not cooked all the way through, about 3-4 minutes per side. Remove from pan and set aside.
Add bacon to the pan; sauté for 3 to 5 minutes or until bacon fat has mostly rendered.
Add onion and garlic; sauté for an additional 3 to 5 minutes or until bacon is crispy.
Toss in asparagus, peas and mushrooms; sauté for 3 minutes.
Add white wine and bring to a simmer, scraping any brown bits from the bottom of the pan with a wooden spoon.
Add chicken stock and heavy cream; return to a simmer.
Whisk in chèvre.
Return chicken breasts to the skillet and simmer for 15 minutes.
Season to taste with salt and pepper. Top with fresh chopped tarragon to serve.