If roasting the squash, rub the flesh with a little olive oil, place it on a sheet pan with a bit of water and roast it at 400 degrees for 20 - 30 minutes, until a knife easily passes through the flesh. The red kuri squash roasted more quickly than the acorn.
While the squash is roasting, prepare the rest of the ingredients. Grind the oats in a food processor until they are a coarse meal. If you will be using the food processor to puree the soup, no need to clean it yet.
Heat the olive oil in a large saucepan over medium-low heat. Add the chopped onion, garlic, and 1/4 teaspoon of the salt and cook, stirring occasionally, until the onion is glassy and soft, 6 to 8 minutes.
If sauteing the squash (instead of roasting), increase the heat to medium, add the squash, and cook until the pieces shine with a coating of oil, stirring, about 1 minute. Then increase the heat to medium, add the oats, nutmeg, pepper, and pepper flakes and stir until fragrant, about 1 minute. If you roasted the squash, then increase the heat to medium, add the oats, nutmeg, pepper, and pepper flakes and stir until fragrant, about 1 minute. Now add in the roasted squash, the broth, and the remaining 1/2 teaspoon salt and bring to a boil, scraping the bottom of the pan to release any toasted oatmeal bits.
Decrease the heat to maintain a simmer, cover, and cook until the squash is tender (if you didn't roast it) about 8 minutes, or just let it simmer for 5 - 8 minutes to meld all the flavors if you did roast it.
While the soup simmers, make the spicy yogurt topping. Finely grate the lemon half until you have 1/2 teaspoon zest. Squeeze the fruit to get 1 tablespoon of lemon juice. Add the yogurt, lemon juice, zest, salt, and cayenne to a small serving bowl and beat with a fork until smooth. This can be made up to 3 days ahead of time.
To finish, puree the soup with a handheld blender in the saucepan, or in bathces in the food processor (use caution with hot liquids!). If using a food processor, return the mixture to the saucepan and gently rewarm over medium heat until bubbles appear just below the surface, stirring a few times. Add 1/2 teaspoon of the sugar if desired. Taste for salt, pepper, and sugar and adjust. Ladle the soup into deep plates, spoon a generious tablespoon of topping into each, and garnish with parsley. Be sure to swirl the topping into the soup before you dig in.
If not eating the soup right away, let it cool a little and store away in the refrigerator for up to 3 days. Maria says you can also freeze the soup for up to 1 month but not the yogurt topping.