Heat the oil in a large pot (at least 5 quarts) over medium. Add the onion, carrots, and 1/2 tsp. kosher salt and cook, stirring, until onions are tender and lightly golden, 9 to 10 minutes. Add the garlic and cook, stirring, for 1 minute. Add the button mushrooms and another 1/2 tsp. salt. Increase the heat to medium-high and cook, stirring, until mushrooms release their water, about 5 minutes. Pour in the wine, increase the heat to high, and bring to a boil. Cook until the wine reduces by half, about 3 minutes.