Bring a large pot of salted water to a boil. Add spinach and cook until just wilted, 1 to 2 minutes. Drain spinach, then plunge into a bowl of ice water. Drain spinach, squeezing out any water, and finely chop.

In a large bowl, mix the spinach with the 1 teaspoon of salt, the bread crumbs, parmesan, ricotta, 1 tablespoon of olive oil, and the egg until evenly combined. Divide the dough into 14 balls. Working with 1 ball of dough at a time, flatten the ball in the palm of your hand. Add 2 teaspoons of brie and wrap the dough around the cheese. Dust each ball with flour and place on a parchment paper-lined baking sheet. Freeze dough for 45 minutes.

Bring a large pot of water to a boil. Working in 2 batches, add the dumplings to the water and cook until they float, about 5 to 7 minutes. Using a slotted spoon, transfer dumplings to a plate. Save 1⁄3 cup of the water.

Meanwhile, heat remaining 3 tablespoons oil in a 12-inch skillet over medium-high. Add dumplings and cook, turning as needed, until golden, about 6 minutes. Add the butter and sage and cook, basting the dumplings, 2 minutes longer. Transfer the dumplings to a serving platter and add the reserved water and the vinegar to the skillet. Bring it to a boil and reduce the sauce by half, about 8 minutes. Spoon the sauce over the dumplings and garnish with more parmesan.