Toss cornichons and brine, oil, shallot, and ¼ teaspoon table salt together in bowl; set aside.
Toss tomatoes gently with pepper and remaining ½ teaspoon table salt.
Bring cream to boil in large saucepan over medium-high heat.
Reduce heat to medium-low and simmer, whisking occasionally, until cream is reduced to 1½ cups, 15 to 18 minutes.
Off heat, add fontina, Parmesan, and salt and whisk until cheeses are fully melted and sauce is smooth, about 30 seconds.
Immediately transfer cheese sauce to large platter and spread into even layer.
Arrange tomatoes in even layer over cheese sauce.
Drizzle cornichon dressing over tomatoes.
Sprinkle with thyme and flake sea salt.
Serve.
