Brown the Ground Meat - Add 1 teaspoon of olive oil to a large heavy-bottomed Dutch oven or saucepot set over medium-high heat. Brown the ground beef and sprinkle with salt and pepper, to season. Once cooked, transfer beef to a clean plate and set aside.
Caramelize the Onions (optional) - Set a large, heavy-bottomed Dutch oven over medium heat. Add 1 tablespoon olive oil and 1 tablespoon butter and heat until shimmering and hot. Add the onions, mixing well to coat in oil. Allow your onions to cook, stirring occasionally. If your onions start to brown, reduce heat to medium-low (remember: we're cooking these onions low and slow). After approximately 10 minutes, sprinkle with salt (approximately 1/2 teaspoon) and stir to combine. By now, your onions should be softening and lightly golden. If your onions start to dry out and brown/burn prematurely, as a result, (assuming your heat is set to medium-low), add a few teaspoons of water to the pan.
Continue to cook your onions for an additional 30 minutes or so, stirring every 2-3 minutes to prevent burning. You don't want to stir continuously, but just often enough that the onions have enough time to cook and stick to the bottom of your skillet, browning just slightly - but not burning.
After the first 30 minutes, your onions should be a nice and deep golden color. You may stop now, or continue. If you continue, every minute or so, use a metal spatula to scrape up the brown bits from the bottom of your skillet. Continue to cook, stir, and scrape the brown bits from the bottom of the pan until your onions are a rich golden brown color.
Remove onions from the pot and transfer to a clean cutting board. Roughly chop into smaller pieces using a sharp knife.
Prepare Sauce - Return the Dutch oven to medium-high heat. Add the remaining 3 tablespoons of butter, caramelized onions, and chopped mushrooms to the pot. Stir well to combine. Cook, stirring often, for 5-8 minutes, or until mushrooms have softened.
Season with salt, pepper, Italian seasoning, and fresh garlic. Continue to cook, stirring often, for an additional minute. IF your mixture appears dry, mix an additional 1-2 tablespoons of butter with the onions and mushrooms.
Reduce heat to medium, sprinkle with the all-purpose flour, and immediately mix to combine. Continue mixing for 1 minute, taking extra care not to let the flour burn.
Add the soy sauce, hot sauce, and Worcestershire sauce with the onions and mushrooms. Combine. Stirring continuously, slowly add the milk (I used 2%). Stir until fully combined and heated through - but not boiling. Finally, whisk in the Greek yogurt.
Optional - If you prefer a less chunky sauce or you have little kids who you're worried may not like the bits of onion or mushroom (this would be my kid), simply blend part, or all of, your sauce. I blended HALF with great results.
Assemble - Preheat oven to 375°F and spray a large 9x13 inch baking dish with non-stick cooking spray. Set aside. Add the corn, green beans, and browned ground meat to the sauce. Mix well to combine. Remove from heat and stir in half of the cheese. Transfer the sauce and vegetable mixture to the prepared baking dish. Top with tater tots and sprinkle with remaining cheddar cheese. Transfer to the oven and bake for 30 minutes, or until tater tots are golden brown and crispy. Serve immediately or enjoy leftovers for up to 5 days when stored in the refrigerator.