Arrange the potatoes and bell peppers on a baking sheet. Put the chicken breast on a sheet of foil and fold up the sides to make a boat. Drizzle the chicken with olive oil and sprinkle with salt and black pepper. Put on the baking sheet next to the potatoes and bell peppers.
Bake until the chicken registers 165 degrees F on an instant-read thermometer, about 25 minutes.
Remove the chicken, set aside to cool and then finely chop. Continue to bake the bell peppers and potatoes, turning occasionally, until the bell peppers are soft and their skins wrinkly, about 5 minutes more (about 30 minutes total) and until a knife slips easily into the potatoes, about 35 minutes more (about 1 hour total). When the bell peppers are done, put in a bowl, cover and let steam for 10 minutes. Remove the skins and seeds under cold running water and then roughly chop and set aside.
When the potatoes are done and cool enough to handle, remove the skins and let cool completely.
For the red pepper coulis: Heat 1 tablespoon of the olive oil in medium skillet over medium heat. Add half of the onion and cook, stirring, until soft, about 5 minutes. Add the vinegar and half the garlic and cook, stirring, for 1 minute. Add the chopped bell peppers, chicken broth, 2 tablespoons of the cream cheese, 1/2 teaspoon salt and a few grinds of black pepper. Bring to a simmer and cook until reduced by about half, about 5 minutes. Transfer to a food processor and process until smooth. Transfer to a small bowl and set side. Wipe out the skillet.
For the chicken filling: Heat the remaining tablespoon olive oil in the skillet over medium heat. Add the remaining onion and cook, stirring, until soft, about 5 minutes. Add the spinach and remaining garlic and cook, stirring, until the spinach is completely wilted, about 3 minutes. Add the parsley, chopped chicken, a heaping 1/4 cup cream cheese, 1/2 teaspoon salt and a few grinds of black pepper and stir until smooth and creamy. Set aside to cool.
Divide the filling into 12 even mounds and put on a baking sheet.
For the potato dough: Shred the cooled potatoes on a box grater over a large bowl. Add the remaining cream cheese and 2 teaspoons salt. Use your hands to work into a dough (if the dough is sticky, dip your hands into cold water). Divide into 12 even mounds. Roll each mound into a ball.
Line a baking sheet or large plate with parchment. Split a ball of potato dough in half. Make a deep pocket in one of the halves and stuff it with a mound of chicken filling. Flatten the other ball half with your palm, cover the filling with it and then pinch and roll until the dough is smooth and sealed. Put on the prepared baking sheet and repeat with the remaining dough and filling to make 12 balls. Transfer to the prepared baking sheet and freeze until the balls are chilled and firm, about 20 minutes.
Fill a large pot with 3 inches of vegetable oil and bring to 360 degrees F on a deep-fry thermometer over medium heat. Beat the eggs in shallow bowl, put the flour in another shallow bowl and the panko in a third shallow bowl. Add a large pinch of salt to each and whisk. Roll a potato ball in the flour, then dip in the egg, letting the excess drip off, and then coat in the panko. Return to the baking sheet and bread the remaining balls.
Working in two batches, fry the balls, turning as needed, until golden brown, about 5 minutes. Remove to a paper towel-lined plate and sprinkle with salt.
Spread the coulis on a platter, add the fried balls and garnish with the spinach.