Heat oil in a large Dutch oven or heavy pot over medium heat. Add red onion, chile, and 1 tsp. Diamond Crystal kosher salt (or ½ tsp. Morton kosher salt). Stir with a wooden spoon and cook until the vegetables begin to soften, about 3 minutes.
Stir in garlic and tomato paste, breaking up the paste to coat the vegetables. Then add the squash, potatoes, and broth. Increase the heat to medium-high and bring the broth to a simmer.
Once simmering, reduce heat to medium and cook for 15–20 minutes, or until the potatoes are nearly falling apart.
Add the cherry tomatoes and cook for another 3 minutes until they are warmed through but not bursting. Stir in oregano and vinegar, and let the soup simmer for 2 minutes to meld the flavors together.
Remove from heat and stir in the milk (if using). Taste the soup and adjust seasoning with more salt if needed.
To serve, divide the queso fresco between bowls and ladle the soup over the cheese, stirring gently to prevent clumping.
Enjoy!