Set a 10-inch skillet over medium heat, about 5 on a 10-dial, and add the extra-virgin olive oil. Add the 1/4-inch squash cubes and spread them into one loose layer. Cook for 6 minutes and stir every 90 seconds. Then fold in baby spinach, sun-dried tomatoes and thyme. Cook 2 to 4 minutes more. The squash should look pale golden at the edges and dry across the cut faces. The spinach turns deep green and the tomatoes look glossy. You'll smell thyme just before the pan is ready. Go past 10 minutes and the squash can turn mushy. Stop while the cubes still hold shape. Dry sautéing removes water before the eggs set.

Transfer the hot vegetable mixture to a wide plate and cool it for 5 minutes, until it falls below 110°F. Meanwhile, whisk the eggs and whole milk for 45 seconds. Use quick circles and lift through the center every few strokes. Add sea salt and black pepper. Then fold in feta, sharp cheddar and 1 tablespoon parsley. The eggs should look pale yellow with small bubbles and no streaks of white. The cooled vegetables stay distinct instead of melting into the mixture. Fold them in for 20 seconds, just until evenly suspended. Overmixing crushes the squash and makes the custard foamy. Cooling first prevents the hot vegetables from scrambling the eggs.

Keep the oven at 375°F and line all 12 muffin cups with tan parchment liners. Coat each liner with avocado oil spray. Divide the egg mixture within 5 minutes. Scoop from the bowl's bottom each time so every cup gets squash and spinach. Fill each cup three-quarters full. Tap the tin twice against a folded towel to release large air pockets. The 12 portions should look uneven but balanced, with green ribbons and red tomato flecks visible near each surface. No cup should reach the rim. If the bowl sits longer than 5 minutes, the vegetables sink and the last muffins become heavy. Prompt portioning keeps solids evenly suspended through the custard.

Bake the filled tin at 375°F for 18 to 22 minutes on the center rack. Rotate the tin 180 degrees at minute 12. Check the center muffins with an instant-read thermometer after minute 18. Remove the tin when the centers reach 165°F. (My first batch baked at 400°F, the tops set before the centers. 375°F fixed it.) The tops should puff slightly and show irregular yellow-gold patches with amber edges. Feta stays white while sharp cheddar forms tiny toasted spots. You'll smell toasted cheese when the center is set. A thin knife should leave only moist crumbs. Bake past 22 minutes and the eggs turn rubbery. Moderate heat sets the proteins before their moisture escapes.

Rest the tin at room temperature for exactly 5 minutes. Run a thin silicone spatula around each muffin and lift gently from opposite sides. Move one muffin to a plate and cut it in half after 2 more minutes. Scatter the reserved parsley over the cut halves. Serve while the center stays above 140°F. The cut center should look moist but fully set, with soft squash cubes and clear feta pockets. The edges stay amber and slightly springy. If you unmold before 5 minutes, the tender sides can tear. Wait longer than 15 minutes and steam softens the edges. Brief resting lets the egg structure settle before cutting.