Heat olive oil in a large oven-safe skillet over medium heat. Add sliced shallots and cook, stirring occasionally, until deeply golden brown and caramelized, about 15-20 minutes. Remove shallots from skillet and set aside on a paper towel-lined plate.

Add butter to the same skillet over medium-high heat. Once melted, add mushrooms and cook without stirring for 3-4 minutes to allow them to brown. Stir and continue cooking until mushrooms are tender and nicely browned, about 8-10 minutes more. Season with salt and pepper.

Add trimmed green beans and minced garlic to the skillet with the mushrooms. Cook, stirring frequently, until the green beans are bright green and tender-crisp, about 5-7 minutes.

Sprinkle flour over the vegetables in the skillet. Stir continuously for 1-2 minutes to cook out the raw flour taste, coating the vegetables.

Gradually whisk in the vegetable broth, scraping up any browned bits from the bottom of the skillet. Bring to a simmer and cook until the sauce begins to thicken.

Stir in the heavy cream, soy sauce, Worcestershire sauce, and fresh thyme. Continue to simmer gently until the sauce is rich and coats the back of a spoon, about 3-5 minutes. Taste and adjust seasoning with salt and pepper.

Preheat your oven broiler. Sprinkle the reserved crispy fried shallots evenly over the top of the green bean and mushroom mixture. Broil for 2-4 minutes, watching carefully, until the shallots are golden and bubbly.

Serve immediately directly from the skillet.