In a large bowl, squeeze excess liquid from the thawed spinach until mostly dry. Add the ricotta, egg yolks, 1/2 cup flour, 3/4 cup grated Parmesan, and 1 teaspoon salt (use the pinch listed in ingredients as a starting point). Stir until just combined; the dough should be sticky. If it feels too wet to form, add flour 1 tablespoon at a time until it holds together.
Flour a work surface and turn the dough out onto it. Divide the dough into four pieces. Working with one piece at a time, roll into a rope about 1 inch thick, then cut into bite-size pieces. Place the gnocchi on a floured baking sheet and repeat with remaining dough.
Bring a large pot of salted water to a boil. Cook the gnocchi in batches until they float and are cooked through, about 3–4 minutes. Remove with a slotted spoon and set aside while you make the sauces.
In a large skillet over medium heat, melt the butter with the smashed garlic and chopped sage. Cook, stirring occasionally, until the butter browns and the sage becomes crisp, about 4–5 minutes. Remove and discard the garlic if desired.
Add the cooked gnocchi to the brown butter and gently toss to coat. Keep over low heat to stay warm while you finish the tomatoes.
In a separate skillet over medium heat, warm the olive oil. Add the cherry tomatoes and a pinch of salt and pepper. Cook, stirring occasionally, until the tomatoes begin to blister and pop, about 4–5 minutes. Turn off the heat and toss with the fresh thyme and balsamic vinegar.
Divide the gnocchi among bowls, spoon the sage butter over top, then scatter the blistered cherry tomatoes and any pan juices. Sprinkle additional Parmesan if desired and serve immediately.