Why does the De Buyer pan cause food to burn

You've got a great pan and in a short time I'm sure you'll come to love it.

When using a standard pan (one without non-stick coating), heat your pan dry over high heat until you can hold your hand about 6-inches above the cooking surface and feel the heat radiating upward. This allows the tiny cracks and crevices that are imperceptible to the bare hand to expand and when the oil is added, it will coat and create a more even cooking surface.

Add just enough oil to lightly coat the surface. Adding to much oil leads to pan-frying which is fine if you're frying chicken but not what you want when searing and sauteing meat. There should just be a thin film across the bottom of the pan. An additional benefit to first heating the pan is the fact that it will actually take less oil to coat the pan due to the decreased viscosity. When the oil hits the hot pan it will instantly heat and should shimmer across the bottom like water on a freshly cleaned windshield.

You don't put the oil in the pan first because the longer oils and fats heat the quicker they break down and smoke. If you were to add cold oil and cold food to a cold pan and then start heating, you just end up with a big sticky mess.

Make sure you're prepared to add the food to the pan once the oil goes in otherwise the oil will start to burn.

The issue with burning and overcooking is going to be a matter of controlling the heat. Start searing and sauteeing over high heat because as food is added it will suck a good deal of heat from the pan. If it isn't extremely hot to begin with you'll end up with a steaming mess of gray colored meat or vegetables that aren't doing much cooking. Once the meat is browned but needs further cooking you can always turn down the heat to prevent excessive browning and crusting before the interior is done.

Even if you aren't planning to do a pan sauce, or if you've burnt what was cooking in the pan, you'll still want to deglaze with some water while the pan is hot (you can reheat it if it has already cooled down) so that you can scraped up the cooked on bits more easily and have less scrubbing to do when cleaning the pan.

First, about the temperature. Your safest option is to use a gun ;) The correct temp for shallow frying is between 150 and 190 degrees celsius. So if you have an infrared "gun" (a thermometer which neasures the temp of the surface at which it is pointed), use it to determine the stove setting at which the temperature of the dry (not ptfe coated) pan stabilizes in this interval after longer heating. If you don't have it, use the typical tricks for determining the pan temp, they are described in other questions here on SA.

Important: different stoves heat to different temperatures. Don't let the numbers at the knob to fool it into thinking that a given temp is "medium high" because it has the number 7. My own stove stabilizes at ~180 degrees C at setting 2 (out of 10), with setting 3 pushing 230. On the other hand, the minimum temperature for cooking fish is 50 deg. As stoves rarely go below 80 degrees, there is practically no chance to set your stove to a temperature so low that the fish stays undercooked (but it might need unreasonably long times on very low settings).

As for the spraying: the fat at frying temperature will always spray. You can still reduce your problems. First, the bubble explosions at the lower temperature will be less violent. Second, surface moisture promotes spraying (which is produced by water being instantly turned into steam in the oil). So pat the fish dry with paper before frying it. If this isn't enough, you can also drag it through flour. This will make the crust crisper. As the spraying won't be eliminated completely, don't throw in the pieces. Take a flat spatula with a long handle, place the piece on the working end, step back from the stove and slowly lay the fish into the oil, gradually pulling the spatula from beneath it. The turning should produce less spraying, so using a fork or spoon to press the fish against the spatula isn't a problem (or kitchen pincers, if you have those).

All of the above applies to everything you are shallow frying, not only fish. Only the flour is unsuitable for some vegetables.