In a large mixing bowl, combine the warm milk and honey. Sprinkle the instant dry yeast over the top and let it sit for about 5 minutes until it becomes frothy.
Add the beaten eggs and melted butter to the yeast mixture. Stir until well combined.
Gradually add 3 cups of all-purpose flour and the kosher salt to the wet ingredients. Mix with a wooden spoon until a dough begins to form. If the dough is too sticky, add more flour, a little at a time, until it is manageable.
Turn the dough onto a floured surface and knead for about 5-7 minutes, or until it is smooth and elastic. Form the dough into a ball.
Place the dough in a lightly greased bowl and cover it with a kitchen towel or plastic wrap. Let it rise in a warm place for about 1 hour or until it doubles in size.
While the dough is rising, mix the shredded cheddar cheese, thyme leaves, everything bagel spice, and crushed red pepper flakes in a bowl. Set aside.
Once the dough has risen, punch it down gently and turn it out onto a floured surface. Roll the dough into a rectangle, approximately 12x18 inches in size.
Spread the basil pesto evenly over the rolled-out dough. Then, sprinkle the cheese mixture on top, making sure to cover the entire surface.
Starting from one long side, carefully roll the dough into a tight log. Pinch the seams to seal.
Using a sharp knife, slice the log into 12 equal pieces. Place the rolls cut side up on a parchment-lined baking sheet.
Cover the rolls with a kitchen towel and let them rise again for about 30 minutes, until they puff up.
Preheat your oven to 375°F (190°C). Once the rolls have risen, bake them for 20-25 minutes or until they are golden brown on top and the cheese is bubbling.
Remove the buns from the oven and let them cool slightly before serving. Enjoy them warm for the best flavor and texture!