Place an oven rack in the center position and preheat your oven to 375°F (190°C).
Peel and finely dice the half onion, dice the red bell pepper into 1/4-inch pieces, mince the garlic, and rinse and dry the baby spinach. Grate the cheddar and Parmigiano Reggiano, and chop the chives.
Crack all 8 eggs into a large mixing bowl, add the 1/2 cup milk, 1/4 teaspoon kosher salt, and 1/4 teaspoon black pepper. Whisk for about 30 seconds, just until combined and slightly frothy. Stir in the 1/4 cup Parmigiano Reggiano, 1/2 cup cheddar, and 1 tablespoon chopped chives.
Place a 10-inch oven-safe skillet over medium heat and add the olive oil and butter. Once the butter melts and foams, add the diced onion, diced bell pepper, and remaining 1/4 teaspoon kosher salt. Cook for 5 to 6 minutes, stirring often, until the onion is soft and translucent and the pepper is tender.
Stir in the minced garlic and cook for about 30 seconds, just until fragrant. Add the spinach in two or three handfuls, stirring after each one, and cook for 1 to 2 minutes until wilted and any liquid has evaporated. Spread the vegetables into an even layer.
Reduce the heat to medium-low. Give the egg mixture a quick final whisk and pour it evenly over the vegetables, making sure it reaches all the way to the edges. Gently stir once or twice, then leave it alone. Cook for 3 to 4 minutes, until the edges look set and slightly pulled away from the sides, while the center is still loose and jiggly.
Sprinkle the remaining 1/4 cup shredded cheddar evenly over the surface. Carefully transfer the skillet to the center rack of the preheated oven and bake for 12 to 15 minutes, until the frittata is puffed, the top is lightly golden, and the center no longer looks wet. The center should read about 160°F on an instant-read thermometer.
Remove the skillet from the oven using oven mitts and place it on a trivet or heatproof surface. Let the frittata rest for 5 minutes. Run a silicone spatula around the edge to loosen it, then sprinkle the remaining tablespoon of chives over the top.
Cut the frittata into 6 wedges and lift each one out with a wide spatula or pie server. Serve warm, or let it cool and enjoy it at room temperature.