Warm a large Dutch oven or stockpot over medium heat, then cook the bacon for about 5 minutes until crisp. The bacon should sizzle steadily and smell deeply savory, with browned edges and rendered fat in the pot.
Stir in the onion and celery and cook for 3 minutes, then add the corn and cook 4 minutes more before adding the garlic for about 1 minute. The onion should look translucent, the corn should brighten in color, and the garlic should smell fragrant without turning dark.
Pour in the chicken broth and raise the heat to medium-high; when the liquid reaches a simmer, add the potatoes, salt, pepper, paprika, parsley, thyme, and cayenne, then lower the heat to medium and cook for 10 minutes. You want a steady simmer, not a hard boil, and the potatoes should begin to soften around the edges.
Add the zucchini and yellow squash and continue cooking for 10 to 12 minutes. The potatoes should be fork-tender before you move on, while the squash should be cooked through but still hold its shape.
Scoop 2 cups of the chowder into a blender or food processor and puree for 1 to 2 minutes until smooth, then return it to the pot and stir in the half and half before taking the pot off the heat. Once the puree goes back in, the soup should look noticeably creamier and a little thicker without needing flour.
Let the chowder rest for 10 minutes before serving. This short pause helps the broth settle and the texture become slightly fuller in the bowl.